Properties

Label 507.2.a.f.1.1
Level $507$
Weight $2$
Character 507.1
Self dual yes
Analytic conductor $4.048$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $2$

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Newspace parameters

Level: \( N \) \(=\) \( 507 = 3 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 507.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.04841538248\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \(x^{2} - 3\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 39)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 507.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -3.46410 q^{7} +1.73205 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.73205 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.73205 q^{6} -3.46410 q^{7} +1.73205 q^{8} +1.00000 q^{9} -3.46410 q^{11} -1.00000 q^{12} +6.00000 q^{14} -5.00000 q^{16} +6.00000 q^{17} -1.73205 q^{18} -3.46410 q^{19} +3.46410 q^{21} +6.00000 q^{22} -1.73205 q^{24} -5.00000 q^{25} -1.00000 q^{27} -3.46410 q^{28} +6.00000 q^{29} +3.46410 q^{31} +5.19615 q^{32} +3.46410 q^{33} -10.3923 q^{34} +1.00000 q^{36} +6.92820 q^{37} +6.00000 q^{38} +6.92820 q^{41} -6.00000 q^{42} +4.00000 q^{43} -3.46410 q^{44} +3.46410 q^{47} +5.00000 q^{48} +5.00000 q^{49} +8.66025 q^{50} -6.00000 q^{51} +6.00000 q^{53} +1.73205 q^{54} -6.00000 q^{56} +3.46410 q^{57} -10.3923 q^{58} +10.3923 q^{59} -2.00000 q^{61} -6.00000 q^{62} -3.46410 q^{63} +1.00000 q^{64} -6.00000 q^{66} +10.3923 q^{67} +6.00000 q^{68} -3.46410 q^{71} +1.73205 q^{72} -12.0000 q^{74} +5.00000 q^{75} -3.46410 q^{76} +12.0000 q^{77} -8.00000 q^{79} +1.00000 q^{81} -12.0000 q^{82} +3.46410 q^{83} +3.46410 q^{84} -6.92820 q^{86} -6.00000 q^{87} -6.00000 q^{88} -6.92820 q^{89} -3.46410 q^{93} -6.00000 q^{94} -5.19615 q^{96} +13.8564 q^{97} -8.66025 q^{98} -3.46410 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{3} + 2q^{4} + 2q^{9} + O(q^{10}) \) \( 2q - 2q^{3} + 2q^{4} + 2q^{9} - 2q^{12} + 12q^{14} - 10q^{16} + 12q^{17} + 12q^{22} - 10q^{25} - 2q^{27} + 12q^{29} + 2q^{36} + 12q^{38} - 12q^{42} + 8q^{43} + 10q^{48} + 10q^{49} - 12q^{51} + 12q^{53} - 12q^{56} - 4q^{61} - 12q^{62} + 2q^{64} - 12q^{66} + 12q^{68} - 24q^{74} + 10q^{75} + 24q^{77} - 16q^{79} + 2q^{81} - 24q^{82} - 12q^{87} - 12q^{88} - 12q^{94} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(6\) 1.73205 0.707107
\(7\) −3.46410 −1.30931 −0.654654 0.755929i \(-0.727186\pi\)
−0.654654 + 0.755929i \(0.727186\pi\)
\(8\) 1.73205 0.612372
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −3.46410 −1.04447 −0.522233 0.852803i \(-0.674901\pi\)
−0.522233 + 0.852803i \(0.674901\pi\)
\(12\) −1.00000 −0.288675
\(13\) 0 0
\(14\) 6.00000 1.60357
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) 6.00000 1.45521 0.727607 0.685994i \(-0.240633\pi\)
0.727607 + 0.685994i \(0.240633\pi\)
\(18\) −1.73205 −0.408248
\(19\) −3.46410 −0.794719 −0.397360 0.917663i \(-0.630073\pi\)
−0.397360 + 0.917663i \(0.630073\pi\)
\(20\) 0 0
\(21\) 3.46410 0.755929
\(22\) 6.00000 1.27920
\(23\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(24\) −1.73205 −0.353553
\(25\) −5.00000 −1.00000
\(26\) 0 0
\(27\) −1.00000 −0.192450
\(28\) −3.46410 −0.654654
\(29\) 6.00000 1.11417 0.557086 0.830455i \(-0.311919\pi\)
0.557086 + 0.830455i \(0.311919\pi\)
\(30\) 0 0
\(31\) 3.46410 0.622171 0.311086 0.950382i \(-0.399307\pi\)
0.311086 + 0.950382i \(0.399307\pi\)
\(32\) 5.19615 0.918559
\(33\) 3.46410 0.603023
\(34\) −10.3923 −1.78227
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) 6.92820 1.13899 0.569495 0.821995i \(-0.307139\pi\)
0.569495 + 0.821995i \(0.307139\pi\)
\(38\) 6.00000 0.973329
\(39\) 0 0
\(40\) 0 0
\(41\) 6.92820 1.08200 0.541002 0.841021i \(-0.318045\pi\)
0.541002 + 0.841021i \(0.318045\pi\)
\(42\) −6.00000 −0.925820
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) −3.46410 −0.522233
\(45\) 0 0
\(46\) 0 0
\(47\) 3.46410 0.505291 0.252646 0.967559i \(-0.418699\pi\)
0.252646 + 0.967559i \(0.418699\pi\)
\(48\) 5.00000 0.721688
\(49\) 5.00000 0.714286
\(50\) 8.66025 1.22474
\(51\) −6.00000 −0.840168
\(52\) 0 0
\(53\) 6.00000 0.824163 0.412082 0.911147i \(-0.364802\pi\)
0.412082 + 0.911147i \(0.364802\pi\)
\(54\) 1.73205 0.235702
\(55\) 0 0
\(56\) −6.00000 −0.801784
\(57\) 3.46410 0.458831
\(58\) −10.3923 −1.36458
\(59\) 10.3923 1.35296 0.676481 0.736460i \(-0.263504\pi\)
0.676481 + 0.736460i \(0.263504\pi\)
\(60\) 0 0
\(61\) −2.00000 −0.256074 −0.128037 0.991769i \(-0.540868\pi\)
−0.128037 + 0.991769i \(0.540868\pi\)
\(62\) −6.00000 −0.762001
\(63\) −3.46410 −0.436436
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) −6.00000 −0.738549
\(67\) 10.3923 1.26962 0.634811 0.772667i \(-0.281078\pi\)
0.634811 + 0.772667i \(0.281078\pi\)
\(68\) 6.00000 0.727607
\(69\) 0 0
\(70\) 0 0
\(71\) −3.46410 −0.411113 −0.205557 0.978645i \(-0.565900\pi\)
−0.205557 + 0.978645i \(0.565900\pi\)
\(72\) 1.73205 0.204124
\(73\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(74\) −12.0000 −1.39497
\(75\) 5.00000 0.577350
\(76\) −3.46410 −0.397360
\(77\) 12.0000 1.36753
\(78\) 0 0
\(79\) −8.00000 −0.900070 −0.450035 0.893011i \(-0.648589\pi\)
−0.450035 + 0.893011i \(0.648589\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) −12.0000 −1.32518
\(83\) 3.46410 0.380235 0.190117 0.981761i \(-0.439113\pi\)
0.190117 + 0.981761i \(0.439113\pi\)
\(84\) 3.46410 0.377964
\(85\) 0 0
\(86\) −6.92820 −0.747087
\(87\) −6.00000 −0.643268
\(88\) −6.00000 −0.639602
\(89\) −6.92820 −0.734388 −0.367194 0.930144i \(-0.619682\pi\)
−0.367194 + 0.930144i \(0.619682\pi\)
\(90\) 0 0
\(91\) 0 0
\(92\) 0 0
\(93\) −3.46410 −0.359211
\(94\) −6.00000 −0.618853
\(95\) 0 0
\(96\) −5.19615 −0.530330
\(97\) 13.8564 1.40690 0.703452 0.710742i \(-0.251641\pi\)
0.703452 + 0.710742i \(0.251641\pi\)
\(98\) −8.66025 −0.874818
\(99\) −3.46410 −0.348155
\(100\) −5.00000 −0.500000
\(101\) −6.00000 −0.597022 −0.298511 0.954406i \(-0.596490\pi\)
−0.298511 + 0.954406i \(0.596490\pi\)
\(102\) 10.3923 1.02899
\(103\) −8.00000 −0.788263 −0.394132 0.919054i \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) 0 0
\(105\) 0 0
\(106\) −10.3923 −1.00939
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.92820 −0.663602 −0.331801 0.943349i \(-0.607656\pi\)
−0.331801 + 0.943349i \(0.607656\pi\)
\(110\) 0 0
\(111\) −6.92820 −0.657596
\(112\) 17.3205 1.63663
\(113\) −6.00000 −0.564433 −0.282216 0.959351i \(-0.591070\pi\)
−0.282216 + 0.959351i \(0.591070\pi\)
\(114\) −6.00000 −0.561951
\(115\) 0 0
\(116\) 6.00000 0.557086
\(117\) 0 0
\(118\) −18.0000 −1.65703
\(119\) −20.7846 −1.90532
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) 3.46410 0.313625
\(123\) −6.92820 −0.624695
\(124\) 3.46410 0.311086
\(125\) 0 0
\(126\) 6.00000 0.534522
\(127\) 8.00000 0.709885 0.354943 0.934888i \(-0.384500\pi\)
0.354943 + 0.934888i \(0.384500\pi\)
\(128\) −12.1244 −1.07165
\(129\) −4.00000 −0.352180
\(130\) 0 0
\(131\) −12.0000 −1.04844 −0.524222 0.851581i \(-0.675644\pi\)
−0.524222 + 0.851581i \(0.675644\pi\)
\(132\) 3.46410 0.301511
\(133\) 12.0000 1.04053
\(134\) −18.0000 −1.55496
\(135\) 0 0
\(136\) 10.3923 0.891133
\(137\) −20.7846 −1.77575 −0.887875 0.460086i \(-0.847819\pi\)
−0.887875 + 0.460086i \(0.847819\pi\)
\(138\) 0 0
\(139\) −4.00000 −0.339276 −0.169638 0.985506i \(-0.554260\pi\)
−0.169638 + 0.985506i \(0.554260\pi\)
\(140\) 0 0
\(141\) −3.46410 −0.291730
\(142\) 6.00000 0.503509
\(143\) 0 0
\(144\) −5.00000 −0.416667
\(145\) 0 0
\(146\) 0 0
\(147\) −5.00000 −0.412393
\(148\) 6.92820 0.569495
\(149\) −13.8564 −1.13516 −0.567581 0.823318i \(-0.692120\pi\)
−0.567581 + 0.823318i \(0.692120\pi\)
\(150\) −8.66025 −0.707107
\(151\) 10.3923 0.845714 0.422857 0.906196i \(-0.361027\pi\)
0.422857 + 0.906196i \(0.361027\pi\)
\(152\) −6.00000 −0.486664
\(153\) 6.00000 0.485071
\(154\) −20.7846 −1.67487
\(155\) 0 0
\(156\) 0 0
\(157\) 14.0000 1.11732 0.558661 0.829396i \(-0.311315\pi\)
0.558661 + 0.829396i \(0.311315\pi\)
\(158\) 13.8564 1.10236
\(159\) −6.00000 −0.475831
\(160\) 0 0
\(161\) 0 0
\(162\) −1.73205 −0.136083
\(163\) 3.46410 0.271329 0.135665 0.990755i \(-0.456683\pi\)
0.135665 + 0.990755i \(0.456683\pi\)
\(164\) 6.92820 0.541002
\(165\) 0 0
\(166\) −6.00000 −0.465690
\(167\) 17.3205 1.34030 0.670151 0.742225i \(-0.266230\pi\)
0.670151 + 0.742225i \(0.266230\pi\)
\(168\) 6.00000 0.462910
\(169\) 0 0
\(170\) 0 0
\(171\) −3.46410 −0.264906
\(172\) 4.00000 0.304997
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 10.3923 0.787839
\(175\) 17.3205 1.30931
\(176\) 17.3205 1.30558
\(177\) −10.3923 −0.781133
\(178\) 12.0000 0.899438
\(179\) 12.0000 0.896922 0.448461 0.893802i \(-0.351972\pi\)
0.448461 + 0.893802i \(0.351972\pi\)
\(180\) 0 0
\(181\) 10.0000 0.743294 0.371647 0.928374i \(-0.378793\pi\)
0.371647 + 0.928374i \(0.378793\pi\)
\(182\) 0 0
\(183\) 2.00000 0.147844
\(184\) 0 0
\(185\) 0 0
\(186\) 6.00000 0.439941
\(187\) −20.7846 −1.51992
\(188\) 3.46410 0.252646
\(189\) 3.46410 0.251976
\(190\) 0 0
\(191\) 24.0000 1.73658 0.868290 0.496058i \(-0.165220\pi\)
0.868290 + 0.496058i \(0.165220\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(194\) −24.0000 −1.72310
\(195\) 0 0
\(196\) 5.00000 0.357143
\(197\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(198\) 6.00000 0.426401
\(199\) −16.0000 −1.13421 −0.567105 0.823646i \(-0.691937\pi\)
−0.567105 + 0.823646i \(0.691937\pi\)
\(200\) −8.66025 −0.612372
\(201\) −10.3923 −0.733017
\(202\) 10.3923 0.731200
\(203\) −20.7846 −1.45879
\(204\) −6.00000 −0.420084
\(205\) 0 0
\(206\) 13.8564 0.965422
\(207\) 0 0
\(208\) 0 0
\(209\) 12.0000 0.830057
\(210\) 0 0
\(211\) −20.0000 −1.37686 −0.688428 0.725304i \(-0.741699\pi\)
−0.688428 + 0.725304i \(0.741699\pi\)
\(212\) 6.00000 0.412082
\(213\) 3.46410 0.237356
\(214\) −20.7846 −1.42081
\(215\) 0 0
\(216\) −1.73205 −0.117851
\(217\) −12.0000 −0.814613
\(218\) 12.0000 0.812743
\(219\) 0 0
\(220\) 0 0
\(221\) 0 0
\(222\) 12.0000 0.805387
\(223\) 3.46410 0.231973 0.115987 0.993251i \(-0.462997\pi\)
0.115987 + 0.993251i \(0.462997\pi\)
\(224\) −18.0000 −1.20268
\(225\) −5.00000 −0.333333
\(226\) 10.3923 0.691286
\(227\) 17.3205 1.14960 0.574801 0.818293i \(-0.305079\pi\)
0.574801 + 0.818293i \(0.305079\pi\)
\(228\) 3.46410 0.229416
\(229\) −6.92820 −0.457829 −0.228914 0.973447i \(-0.573518\pi\)
−0.228914 + 0.973447i \(0.573518\pi\)
\(230\) 0 0
\(231\) −12.0000 −0.789542
\(232\) 10.3923 0.682288
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) 0 0
\(236\) 10.3923 0.676481
\(237\) 8.00000 0.519656
\(238\) 36.0000 2.33353
\(239\) 10.3923 0.672222 0.336111 0.941822i \(-0.390888\pi\)
0.336111 + 0.941822i \(0.390888\pi\)
\(240\) 0 0
\(241\) 13.8564 0.892570 0.446285 0.894891i \(-0.352747\pi\)
0.446285 + 0.894891i \(0.352747\pi\)
\(242\) −1.73205 −0.111340
\(243\) −1.00000 −0.0641500
\(244\) −2.00000 −0.128037
\(245\) 0 0
\(246\) 12.0000 0.765092
\(247\) 0 0
\(248\) 6.00000 0.381000
\(249\) −3.46410 −0.219529
\(250\) 0 0
\(251\) 12.0000 0.757433 0.378717 0.925513i \(-0.376365\pi\)
0.378717 + 0.925513i \(0.376365\pi\)
\(252\) −3.46410 −0.218218
\(253\) 0 0
\(254\) −13.8564 −0.869428
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) −18.0000 −1.12281 −0.561405 0.827541i \(-0.689739\pi\)
−0.561405 + 0.827541i \(0.689739\pi\)
\(258\) 6.92820 0.431331
\(259\) −24.0000 −1.49129
\(260\) 0 0
\(261\) 6.00000 0.371391
\(262\) 20.7846 1.28408
\(263\) −24.0000 −1.47990 −0.739952 0.672660i \(-0.765152\pi\)
−0.739952 + 0.672660i \(0.765152\pi\)
\(264\) 6.00000 0.369274
\(265\) 0 0
\(266\) −20.7846 −1.27439
\(267\) 6.92820 0.423999
\(268\) 10.3923 0.634811
\(269\) 6.00000 0.365826 0.182913 0.983129i \(-0.441447\pi\)
0.182913 + 0.983129i \(0.441447\pi\)
\(270\) 0 0
\(271\) 10.3923 0.631288 0.315644 0.948878i \(-0.397780\pi\)
0.315644 + 0.948878i \(0.397780\pi\)
\(272\) −30.0000 −1.81902
\(273\) 0 0
\(274\) 36.0000 2.17484
\(275\) 17.3205 1.04447
\(276\) 0 0
\(277\) 10.0000 0.600842 0.300421 0.953807i \(-0.402873\pi\)
0.300421 + 0.953807i \(0.402873\pi\)
\(278\) 6.92820 0.415526
\(279\) 3.46410 0.207390
\(280\) 0 0
\(281\) 6.92820 0.413302 0.206651 0.978415i \(-0.433744\pi\)
0.206651 + 0.978415i \(0.433744\pi\)
\(282\) 6.00000 0.357295
\(283\) −4.00000 −0.237775 −0.118888 0.992908i \(-0.537933\pi\)
−0.118888 + 0.992908i \(0.537933\pi\)
\(284\) −3.46410 −0.205557
\(285\) 0 0
\(286\) 0 0
\(287\) −24.0000 −1.41668
\(288\) 5.19615 0.306186
\(289\) 19.0000 1.11765
\(290\) 0 0
\(291\) −13.8564 −0.812277
\(292\) 0 0
\(293\) 27.7128 1.61900 0.809500 0.587120i \(-0.199738\pi\)
0.809500 + 0.587120i \(0.199738\pi\)
\(294\) 8.66025 0.505076
\(295\) 0 0
\(296\) 12.0000 0.697486
\(297\) 3.46410 0.201008
\(298\) 24.0000 1.39028
\(299\) 0 0
\(300\) 5.00000 0.288675
\(301\) −13.8564 −0.798670
\(302\) −18.0000 −1.03578
\(303\) 6.00000 0.344691
\(304\) 17.3205 0.993399
\(305\) 0 0
\(306\) −10.3923 −0.594089
\(307\) −10.3923 −0.593120 −0.296560 0.955014i \(-0.595840\pi\)
−0.296560 + 0.955014i \(0.595840\pi\)
\(308\) 12.0000 0.683763
\(309\) 8.00000 0.455104
\(310\) 0 0
\(311\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(312\) 0 0
\(313\) 10.0000 0.565233 0.282617 0.959233i \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) −24.2487 −1.36843
\(315\) 0 0
\(316\) −8.00000 −0.450035
\(317\) −13.8564 −0.778253 −0.389127 0.921184i \(-0.627223\pi\)
−0.389127 + 0.921184i \(0.627223\pi\)
\(318\) 10.3923 0.582772
\(319\) −20.7846 −1.16371
\(320\) 0 0
\(321\) −12.0000 −0.669775
\(322\) 0 0
\(323\) −20.7846 −1.15649
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −6.00000 −0.332309
\(327\) 6.92820 0.383131
\(328\) 12.0000 0.662589
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −3.46410 −0.190404 −0.0952021 0.995458i \(-0.530350\pi\)
−0.0952021 + 0.995458i \(0.530350\pi\)
\(332\) 3.46410 0.190117
\(333\) 6.92820 0.379663
\(334\) −30.0000 −1.64153
\(335\) 0 0
\(336\) −17.3205 −0.944911
\(337\) 14.0000 0.762629 0.381314 0.924445i \(-0.375472\pi\)
0.381314 + 0.924445i \(0.375472\pi\)
\(338\) 0 0
\(339\) 6.00000 0.325875
\(340\) 0 0
\(341\) −12.0000 −0.649836
\(342\) 6.00000 0.324443
\(343\) 6.92820 0.374088
\(344\) 6.92820 0.373544
\(345\) 0 0
\(346\) −31.1769 −1.67608
\(347\) 36.0000 1.93258 0.966291 0.257454i \(-0.0828835\pi\)
0.966291 + 0.257454i \(0.0828835\pi\)
\(348\) −6.00000 −0.321634
\(349\) −6.92820 −0.370858 −0.185429 0.982658i \(-0.559368\pi\)
−0.185429 + 0.982658i \(0.559368\pi\)
\(350\) −30.0000 −1.60357
\(351\) 0 0
\(352\) −18.0000 −0.959403
\(353\) −34.6410 −1.84376 −0.921878 0.387481i \(-0.873345\pi\)
−0.921878 + 0.387481i \(0.873345\pi\)
\(354\) 18.0000 0.956689
\(355\) 0 0
\(356\) −6.92820 −0.367194
\(357\) 20.7846 1.10004
\(358\) −20.7846 −1.09850
\(359\) 17.3205 0.914141 0.457071 0.889430i \(-0.348899\pi\)
0.457071 + 0.889430i \(0.348899\pi\)
\(360\) 0 0
\(361\) −7.00000 −0.368421
\(362\) −17.3205 −0.910346
\(363\) −1.00000 −0.0524864
\(364\) 0 0
\(365\) 0 0
\(366\) −3.46410 −0.181071
\(367\) −16.0000 −0.835193 −0.417597 0.908633i \(-0.637127\pi\)
−0.417597 + 0.908633i \(0.637127\pi\)
\(368\) 0 0
\(369\) 6.92820 0.360668
\(370\) 0 0
\(371\) −20.7846 −1.07908
\(372\) −3.46410 −0.179605
\(373\) 22.0000 1.13912 0.569558 0.821951i \(-0.307114\pi\)
0.569558 + 0.821951i \(0.307114\pi\)
\(374\) 36.0000 1.86152
\(375\) 0 0
\(376\) 6.00000 0.309426
\(377\) 0 0
\(378\) −6.00000 −0.308607
\(379\) −17.3205 −0.889695 −0.444847 0.895606i \(-0.646742\pi\)
−0.444847 + 0.895606i \(0.646742\pi\)
\(380\) 0 0
\(381\) −8.00000 −0.409852
\(382\) −41.5692 −2.12687
\(383\) −3.46410 −0.177007 −0.0885037 0.996076i \(-0.528208\pi\)
−0.0885037 + 0.996076i \(0.528208\pi\)
\(384\) 12.1244 0.618718
\(385\) 0 0
\(386\) 0 0
\(387\) 4.00000 0.203331
\(388\) 13.8564 0.703452
\(389\) 18.0000 0.912636 0.456318 0.889817i \(-0.349168\pi\)
0.456318 + 0.889817i \(0.349168\pi\)
\(390\) 0 0
\(391\) 0 0
\(392\) 8.66025 0.437409
\(393\) 12.0000 0.605320
\(394\) 0 0
\(395\) 0 0
\(396\) −3.46410 −0.174078
\(397\) −34.6410 −1.73858 −0.869291 0.494300i \(-0.835424\pi\)
−0.869291 + 0.494300i \(0.835424\pi\)
\(398\) 27.7128 1.38912
\(399\) −12.0000 −0.600751
\(400\) 25.0000 1.25000
\(401\) 6.92820 0.345978 0.172989 0.984924i \(-0.444657\pi\)
0.172989 + 0.984924i \(0.444657\pi\)
\(402\) 18.0000 0.897758
\(403\) 0 0
\(404\) −6.00000 −0.298511
\(405\) 0 0
\(406\) 36.0000 1.78665
\(407\) −24.0000 −1.18964
\(408\) −10.3923 −0.514496
\(409\) −27.7128 −1.37031 −0.685155 0.728397i \(-0.740266\pi\)
−0.685155 + 0.728397i \(0.740266\pi\)
\(410\) 0 0
\(411\) 20.7846 1.02523
\(412\) −8.00000 −0.394132
\(413\) −36.0000 −1.77144
\(414\) 0 0
\(415\) 0 0
\(416\) 0 0
\(417\) 4.00000 0.195881
\(418\) −20.7846 −1.01661
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) 34.6410 1.68830 0.844150 0.536107i \(-0.180106\pi\)
0.844150 + 0.536107i \(0.180106\pi\)
\(422\) 34.6410 1.68630
\(423\) 3.46410 0.168430
\(424\) 10.3923 0.504695
\(425\) −30.0000 −1.45521
\(426\) −6.00000 −0.290701
\(427\) 6.92820 0.335279
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 0 0
\(431\) 24.2487 1.16802 0.584010 0.811747i \(-0.301483\pi\)
0.584010 + 0.811747i \(0.301483\pi\)
\(432\) 5.00000 0.240563
\(433\) −34.0000 −1.63394 −0.816968 0.576683i \(-0.804347\pi\)
−0.816968 + 0.576683i \(0.804347\pi\)
\(434\) 20.7846 0.997693
\(435\) 0 0
\(436\) −6.92820 −0.331801
\(437\) 0 0
\(438\) 0 0
\(439\) −8.00000 −0.381819 −0.190910 0.981608i \(-0.561144\pi\)
−0.190910 + 0.981608i \(0.561144\pi\)
\(440\) 0 0
\(441\) 5.00000 0.238095
\(442\) 0 0
\(443\) −36.0000 −1.71041 −0.855206 0.518289i \(-0.826569\pi\)
−0.855206 + 0.518289i \(0.826569\pi\)
\(444\) −6.92820 −0.328798
\(445\) 0 0
\(446\) −6.00000 −0.284108
\(447\) 13.8564 0.655386
\(448\) −3.46410 −0.163663
\(449\) 6.92820 0.326962 0.163481 0.986546i \(-0.447728\pi\)
0.163481 + 0.986546i \(0.447728\pi\)
\(450\) 8.66025 0.408248
\(451\) −24.0000 −1.13012
\(452\) −6.00000 −0.282216
\(453\) −10.3923 −0.488273
\(454\) −30.0000 −1.40797
\(455\) 0 0
\(456\) 6.00000 0.280976
\(457\) 27.7128 1.29635 0.648175 0.761491i \(-0.275532\pi\)
0.648175 + 0.761491i \(0.275532\pi\)
\(458\) 12.0000 0.560723
\(459\) −6.00000 −0.280056
\(460\) 0 0
\(461\) 13.8564 0.645357 0.322679 0.946509i \(-0.395417\pi\)
0.322679 + 0.946509i \(0.395417\pi\)
\(462\) 20.7846 0.966988
\(463\) −17.3205 −0.804952 −0.402476 0.915430i \(-0.631850\pi\)
−0.402476 + 0.915430i \(0.631850\pi\)
\(464\) −30.0000 −1.39272
\(465\) 0 0
\(466\) −10.3923 −0.481414
\(467\) −12.0000 −0.555294 −0.277647 0.960683i \(-0.589555\pi\)
−0.277647 + 0.960683i \(0.589555\pi\)
\(468\) 0 0
\(469\) −36.0000 −1.66233
\(470\) 0 0
\(471\) −14.0000 −0.645086
\(472\) 18.0000 0.828517
\(473\) −13.8564 −0.637118
\(474\) −13.8564 −0.636446
\(475\) 17.3205 0.794719
\(476\) −20.7846 −0.952661
\(477\) 6.00000 0.274721
\(478\) −18.0000 −0.823301
\(479\) −10.3923 −0.474837 −0.237418 0.971408i \(-0.576301\pi\)
−0.237418 + 0.971408i \(0.576301\pi\)
\(480\) 0 0
\(481\) 0 0
\(482\) −24.0000 −1.09317
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 0 0
\(486\) 1.73205 0.0785674
\(487\) −38.1051 −1.72671 −0.863354 0.504599i \(-0.831640\pi\)
−0.863354 + 0.504599i \(0.831640\pi\)
\(488\) −3.46410 −0.156813
\(489\) −3.46410 −0.156652
\(490\) 0 0
\(491\) 12.0000 0.541552 0.270776 0.962642i \(-0.412720\pi\)
0.270776 + 0.962642i \(0.412720\pi\)
\(492\) −6.92820 −0.312348
\(493\) 36.0000 1.62136
\(494\) 0 0
\(495\) 0 0
\(496\) −17.3205 −0.777714
\(497\) 12.0000 0.538274
\(498\) 6.00000 0.268866
\(499\) 10.3923 0.465223 0.232612 0.972570i \(-0.425273\pi\)
0.232612 + 0.972570i \(0.425273\pi\)
\(500\) 0 0
\(501\) −17.3205 −0.773823
\(502\) −20.7846 −0.927663
\(503\) 24.0000 1.07011 0.535054 0.844818i \(-0.320291\pi\)
0.535054 + 0.844818i \(0.320291\pi\)
\(504\) −6.00000 −0.267261
\(505\) 0 0
\(506\) 0 0
\(507\) 0 0
\(508\) 8.00000 0.354943
\(509\) 41.5692 1.84252 0.921262 0.388943i \(-0.127160\pi\)
0.921262 + 0.388943i \(0.127160\pi\)
\(510\) 0 0
\(511\) 0 0
\(512\) −8.66025 −0.382733
\(513\) 3.46410 0.152944
\(514\) 31.1769 1.37515
\(515\) 0 0
\(516\) −4.00000 −0.176090
\(517\) −12.0000 −0.527759
\(518\) 41.5692 1.82645
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) 18.0000 0.788594 0.394297 0.918983i \(-0.370988\pi\)
0.394297 + 0.918983i \(0.370988\pi\)
\(522\) −10.3923 −0.454859
\(523\) −4.00000 −0.174908 −0.0874539 0.996169i \(-0.527873\pi\)
−0.0874539 + 0.996169i \(0.527873\pi\)
\(524\) −12.0000 −0.524222
\(525\) −17.3205 −0.755929
\(526\) 41.5692 1.81250
\(527\) 20.7846 0.905392
\(528\) −17.3205 −0.753778
\(529\) −23.0000 −1.00000
\(530\) 0 0
\(531\) 10.3923 0.450988
\(532\) 12.0000 0.520266
\(533\) 0 0
\(534\) −12.0000 −0.519291
\(535\) 0 0
\(536\) 18.0000 0.777482
\(537\) −12.0000 −0.517838
\(538\) −10.3923 −0.448044
\(539\) −17.3205 −0.746047
\(540\) 0 0
\(541\) 6.92820 0.297867 0.148933 0.988847i \(-0.452416\pi\)
0.148933 + 0.988847i \(0.452416\pi\)
\(542\) −18.0000 −0.773166
\(543\) −10.0000 −0.429141
\(544\) 31.1769 1.33670
\(545\) 0 0
\(546\) 0 0
\(547\) 20.0000 0.855138 0.427569 0.903983i \(-0.359370\pi\)
0.427569 + 0.903983i \(0.359370\pi\)
\(548\) −20.7846 −0.887875
\(549\) −2.00000 −0.0853579
\(550\) −30.0000 −1.27920
\(551\) −20.7846 −0.885454
\(552\) 0 0
\(553\) 27.7128 1.17847
\(554\) −17.3205 −0.735878
\(555\) 0 0
\(556\) −4.00000 −0.169638
\(557\) 13.8564 0.587115 0.293557 0.955941i \(-0.405161\pi\)
0.293557 + 0.955941i \(0.405161\pi\)
\(558\) −6.00000 −0.254000
\(559\) 0 0
\(560\) 0 0
\(561\) 20.7846 0.877527
\(562\) −12.0000 −0.506189
\(563\) 12.0000 0.505740 0.252870 0.967500i \(-0.418626\pi\)
0.252870 + 0.967500i \(0.418626\pi\)
\(564\) −3.46410 −0.145865
\(565\) 0 0
\(566\) 6.92820 0.291214
\(567\) −3.46410 −0.145479
\(568\) −6.00000 −0.251754
\(569\) 6.00000 0.251533 0.125767 0.992060i \(-0.459861\pi\)
0.125767 + 0.992060i \(0.459861\pi\)
\(570\) 0 0
\(571\) −4.00000 −0.167395 −0.0836974 0.996491i \(-0.526673\pi\)
−0.0836974 + 0.996491i \(0.526673\pi\)
\(572\) 0 0
\(573\) −24.0000 −1.00261
\(574\) 41.5692 1.73507
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(578\) −32.9090 −1.36883
\(579\) 0 0
\(580\) 0 0
\(581\) −12.0000 −0.497844
\(582\) 24.0000 0.994832
\(583\) −20.7846 −0.860811
\(584\) 0 0
\(585\) 0 0
\(586\) −48.0000 −1.98286
\(587\) −10.3923 −0.428936 −0.214468 0.976731i \(-0.568802\pi\)
−0.214468 + 0.976731i \(0.568802\pi\)
\(588\) −5.00000 −0.206197
\(589\) −12.0000 −0.494451
\(590\) 0 0
\(591\) 0 0
\(592\) −34.6410 −1.42374
\(593\) 6.92820 0.284507 0.142254 0.989830i \(-0.454565\pi\)
0.142254 + 0.989830i \(0.454565\pi\)
\(594\) −6.00000 −0.246183
\(595\) 0 0
\(596\) −13.8564 −0.567581
\(597\) 16.0000 0.654836
\(598\) 0 0
\(599\) 24.0000 0.980613 0.490307 0.871550i \(-0.336885\pi\)
0.490307 + 0.871550i \(0.336885\pi\)
\(600\) 8.66025 0.353553
\(601\) 10.0000 0.407909 0.203954 0.978980i \(-0.434621\pi\)
0.203954 + 0.978980i \(0.434621\pi\)
\(602\) 24.0000 0.978167
\(603\) 10.3923 0.423207
\(604\) 10.3923 0.422857
\(605\) 0 0
\(606\) −10.3923 −0.422159
\(607\) 32.0000 1.29884 0.649420 0.760430i \(-0.275012\pi\)
0.649420 + 0.760430i \(0.275012\pi\)
\(608\) −18.0000 −0.729996
\(609\) 20.7846 0.842235
\(610\) 0 0
\(611\) 0 0
\(612\) 6.00000 0.242536
\(613\) 20.7846 0.839482 0.419741 0.907644i \(-0.362121\pi\)
0.419741 + 0.907644i \(0.362121\pi\)
\(614\) 18.0000 0.726421
\(615\) 0 0
\(616\) 20.7846 0.837436
\(617\) 6.92820 0.278919 0.139459 0.990228i \(-0.455464\pi\)
0.139459 + 0.990228i \(0.455464\pi\)
\(618\) −13.8564 −0.557386
\(619\) 31.1769 1.25311 0.626553 0.779379i \(-0.284465\pi\)
0.626553 + 0.779379i \(0.284465\pi\)
\(620\) 0 0
\(621\) 0 0
\(622\) 0 0
\(623\) 24.0000 0.961540
\(624\) 0 0
\(625\) 25.0000 1.00000
\(626\) −17.3205 −0.692267
\(627\) −12.0000 −0.479234
\(628\) 14.0000 0.558661
\(629\) 41.5692 1.65747
\(630\) 0 0
\(631\) 38.1051 1.51694 0.758470 0.651707i \(-0.225947\pi\)
0.758470 + 0.651707i \(0.225947\pi\)
\(632\) −13.8564 −0.551178
\(633\) 20.0000 0.794929
\(634\) 24.0000 0.953162
\(635\) 0 0
\(636\) −6.00000 −0.237915
\(637\) 0 0
\(638\) 36.0000 1.42525
\(639\) −3.46410 −0.137038
\(640\) 0 0
\(641\) 6.00000 0.236986 0.118493 0.992955i \(-0.462194\pi\)
0.118493 + 0.992955i \(0.462194\pi\)
\(642\) 20.7846 0.820303
\(643\) 10.3923 0.409832 0.204916 0.978780i \(-0.434308\pi\)
0.204916 + 0.978780i \(0.434308\pi\)
\(644\) 0 0
\(645\) 0 0
\(646\) 36.0000 1.41640
\(647\) 24.0000 0.943537 0.471769 0.881722i \(-0.343616\pi\)
0.471769 + 0.881722i \(0.343616\pi\)
\(648\) 1.73205 0.0680414
\(649\) −36.0000 −1.41312
\(650\) 0 0
\(651\) 12.0000 0.470317
\(652\) 3.46410 0.135665
\(653\) 6.00000 0.234798 0.117399 0.993085i \(-0.462544\pi\)
0.117399 + 0.993085i \(0.462544\pi\)
\(654\) −12.0000 −0.469237
\(655\) 0 0
\(656\) −34.6410 −1.35250
\(657\) 0 0
\(658\) 20.7846 0.810268
\(659\) 12.0000 0.467454 0.233727 0.972302i \(-0.424908\pi\)
0.233727 + 0.972302i \(0.424908\pi\)
\(660\) 0 0
\(661\) −20.7846 −0.808428 −0.404214 0.914665i \(-0.632455\pi\)
−0.404214 + 0.914665i \(0.632455\pi\)
\(662\) 6.00000 0.233197
\(663\) 0 0
\(664\) 6.00000 0.232845
\(665\) 0 0
\(666\) −12.0000 −0.464991
\(667\) 0 0
\(668\) 17.3205 0.670151
\(669\) −3.46410 −0.133930
\(670\) 0 0
\(671\) 6.92820 0.267460
\(672\) 18.0000 0.694365
\(673\) 46.0000 1.77317 0.886585 0.462566i \(-0.153071\pi\)
0.886585 + 0.462566i \(0.153071\pi\)
\(674\) −24.2487 −0.934025
\(675\) 5.00000 0.192450
\(676\) 0 0
\(677\) 6.00000 0.230599 0.115299 0.993331i \(-0.463217\pi\)
0.115299 + 0.993331i \(0.463217\pi\)
\(678\) −10.3923 −0.399114
\(679\) −48.0000 −1.84207
\(680\) 0 0
\(681\) −17.3205 −0.663723
\(682\) 20.7846 0.795884
\(683\) −31.1769 −1.19295 −0.596476 0.802631i \(-0.703433\pi\)
−0.596476 + 0.802631i \(0.703433\pi\)
\(684\) −3.46410 −0.132453
\(685\) 0 0
\(686\) −12.0000 −0.458162
\(687\) 6.92820 0.264327
\(688\) −20.0000 −0.762493
\(689\) 0 0
\(690\) 0 0
\(691\) −45.0333 −1.71315 −0.856574 0.516024i \(-0.827412\pi\)
−0.856574 + 0.516024i \(0.827412\pi\)
\(692\) 18.0000 0.684257
\(693\) 12.0000 0.455842
\(694\) −62.3538 −2.36692
\(695\) 0 0
\(696\) −10.3923 −0.393919
\(697\) 41.5692 1.57455
\(698\) 12.0000 0.454207
\(699\) −6.00000 −0.226941
\(700\) 17.3205 0.654654
\(701\) 42.0000 1.58632 0.793159 0.609015i \(-0.208435\pi\)
0.793159 + 0.609015i \(0.208435\pi\)
\(702\) 0 0
\(703\) −24.0000 −0.905177
\(704\) −3.46410 −0.130558
\(705\) 0 0
\(706\) 60.0000 2.25813
\(707\) 20.7846 0.781686
\(708\) −10.3923 −0.390567
\(709\) −6.92820 −0.260194 −0.130097 0.991501i \(-0.541529\pi\)
−0.130097 + 0.991501i \(0.541529\pi\)
\(710\) 0 0
\(711\) −8.00000 −0.300023
\(712\) −12.0000 −0.449719
\(713\) 0 0
\(714\) −36.0000 −1.34727
\(715\) 0 0
\(716\) 12.0000 0.448461
\(717\) −10.3923 −0.388108
\(718\) −30.0000 −1.11959
\(719\) −24.0000 −0.895049 −0.447524 0.894272i \(-0.647694\pi\)
−0.447524 + 0.894272i \(0.647694\pi\)
\(720\) 0 0
\(721\) 27.7128 1.03208
\(722\) 12.1244 0.451222
\(723\) −13.8564 −0.515325
\(724\) 10.0000 0.371647
\(725\) −30.0000 −1.11417
\(726\) 1.73205 0.0642824
\(727\) −16.0000 −0.593407 −0.296704 0.954970i \(-0.595887\pi\)
−0.296704 + 0.954970i \(0.595887\pi\)
\(728\) 0 0
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 24.0000 0.887672
\(732\) 2.00000 0.0739221
\(733\) −34.6410 −1.27950 −0.639748 0.768585i \(-0.720961\pi\)
−0.639748 + 0.768585i \(0.720961\pi\)
\(734\) 27.7128 1.02290
\(735\) 0 0
\(736\) 0 0
\(737\) −36.0000 −1.32608
\(738\) −12.0000 −0.441726
\(739\) −38.1051 −1.40172 −0.700860 0.713299i \(-0.747200\pi\)
−0.700860 + 0.713299i \(0.747200\pi\)
\(740\) 0 0
\(741\) 0 0
\(742\) 36.0000 1.32160
\(743\) −3.46410 −0.127086 −0.0635428 0.997979i \(-0.520240\pi\)
−0.0635428 + 0.997979i \(0.520240\pi\)
\(744\) −6.00000 −0.219971
\(745\) 0 0
\(746\) −38.1051 −1.39513
\(747\) 3.46410 0.126745
\(748\) −20.7846 −0.759961
\(749\) −41.5692 −1.51891
\(750\) 0 0
\(751\) −32.0000 −1.16770 −0.583848 0.811863i \(-0.698454\pi\)
−0.583848 + 0.811863i \(0.698454\pi\)
\(752\) −17.3205 −0.631614
\(753\) −12.0000 −0.437304
\(754\) 0 0
\(755\) 0 0
\(756\) 3.46410 0.125988
\(757\) 22.0000 0.799604 0.399802 0.916602i \(-0.369079\pi\)
0.399802 + 0.916602i \(0.369079\pi\)
\(758\) 30.0000 1.08965
\(759\) 0 0
\(760\) 0 0
\(761\) −48.4974 −1.75803 −0.879015 0.476794i \(-0.841799\pi\)
−0.879015 + 0.476794i \(0.841799\pi\)
\(762\) 13.8564 0.501965
\(763\) 24.0000 0.868858
\(764\) 24.0000 0.868290
\(765\) 0 0
\(766\) 6.00000 0.216789
\(767\) 0 0
\(768\) −19.0000 −0.685603
\(769\) −27.7128 −0.999350 −0.499675 0.866213i \(-0.666547\pi\)
−0.499675 + 0.866213i \(0.666547\pi\)
\(770\) 0 0
\(771\) 18.0000 0.648254
\(772\) 0 0
\(773\) 13.8564 0.498380 0.249190 0.968455i \(-0.419836\pi\)
0.249190 + 0.968455i \(0.419836\pi\)
\(774\) −6.92820 −0.249029
\(775\) −17.3205 −0.622171
\(776\) 24.0000 0.861550
\(777\) 24.0000 0.860995
\(778\) −31.1769 −1.11775
\(779\) −24.0000 −0.859889
\(780\) 0 0
\(781\) 12.0000 0.429394
\(782\) 0 0
\(783\) −6.00000 −0.214423
\(784\) −25.0000 −0.892857
\(785\) 0 0
\(786\) −20.7846 −0.741362
\(787\) −10.3923 −0.370446 −0.185223 0.982697i \(-0.559301\pi\)
−0.185223 + 0.982697i \(0.559301\pi\)
\(788\) 0 0
\(789\) 24.0000 0.854423
\(790\) 0 0
\(791\) 20.7846 0.739016
\(792\) −6.00000 −0.213201
\(793\) 0 0
\(794\) 60.0000 2.12932
\(795\) 0 0
\(796\) −16.0000 −0.567105
\(797\) 42.0000 1.48772 0.743858 0.668338i \(-0.232994\pi\)
0.743858 + 0.668338i \(0.232994\pi\)
\(798\) 20.7846 0.735767
\(799\) 20.7846 0.735307
\(800\) −25.9808 −0.918559
\(801\) −6.92820 −0.244796
\(802\) −12.0000 −0.423735
\(803\) 0 0
\(804\) −10.3923 −0.366508
\(805\) 0 0
\(806\) 0 0
\(807\) −6.00000 −0.211210
\(808\) −10.3923 −0.365600
\(809\) −30.0000 −1.05474 −0.527372 0.849635i \(-0.676823\pi\)
−0.527372 + 0.849635i \(0.676823\pi\)
\(810\) 0 0
\(811\) 38.1051 1.33805 0.669026 0.743239i \(-0.266712\pi\)
0.669026 + 0.743239i \(0.266712\pi\)
\(812\) −20.7846 −0.729397
\(813\) −10.3923 −0.364474
\(814\) 41.5692 1.45700
\(815\) 0 0
\(816\) 30.0000 1.05021
\(817\) −13.8564 −0.484774
\(818\) 48.0000 1.67828
\(819\) 0 0
\(820\) 0 0
\(821\) −13.8564 −0.483592 −0.241796 0.970327i \(-0.577736\pi\)
−0.241796 + 0.970327i \(0.577736\pi\)
\(822\) −36.0000 −1.25564
\(823\) 40.0000 1.39431 0.697156 0.716919i \(-0.254448\pi\)
0.697156 + 0.716919i \(0.254448\pi\)
\(824\) −13.8564 −0.482711
\(825\) −17.3205 −0.603023
\(826\) 62.3538 2.16957
\(827\) 24.2487 0.843210 0.421605 0.906780i \(-0.361467\pi\)
0.421605 + 0.906780i \(0.361467\pi\)
\(828\) 0 0
\(829\) 2.00000 0.0694629 0.0347314 0.999397i \(-0.488942\pi\)
0.0347314 + 0.999397i \(0.488942\pi\)
\(830\) 0 0
\(831\) −10.0000 −0.346896
\(832\) 0 0
\(833\) 30.0000 1.03944
\(834\) −6.92820 −0.239904
\(835\) 0 0
\(836\) 12.0000 0.415029
\(837\) −3.46410 −0.119737
\(838\) 20.7846 0.717992
\(839\) 3.46410 0.119594 0.0597970 0.998211i \(-0.480955\pi\)
0.0597970 + 0.998211i \(0.480955\pi\)
\(840\) 0 0
\(841\) 7.00000 0.241379
\(842\) −60.0000 −2.06774
\(843\) −6.92820 −0.238620
\(844\) −20.0000 −0.688428
\(845\) 0 0
\(846\) −6.00000 −0.206284
\(847\) −3.46410 −0.119028
\(848\) −30.0000 −1.03020
\(849\) 4.00000 0.137280
\(850\) 51.9615 1.78227
\(851\) 0 0
\(852\) 3.46410 0.118678
\(853\) −20.7846 −0.711651 −0.355826 0.934552i \(-0.615800\pi\)
−0.355826 + 0.934552i \(0.615800\pi\)
\(854\) −12.0000 −0.410632
\(855\) 0 0
\(856\) 20.7846 0.710403
\(857\) −42.0000 −1.43469 −0.717346 0.696717i \(-0.754643\pi\)
−0.717346 + 0.696717i \(0.754643\pi\)
\(858\) 0 0
\(859\) 4.00000 0.136478 0.0682391 0.997669i \(-0.478262\pi\)
0.0682391 + 0.997669i \(0.478262\pi\)
\(860\) 0 0
\(861\) 24.0000 0.817918
\(862\) −42.0000 −1.43053
\(863\) −31.1769 −1.06127 −0.530637 0.847599i \(-0.678047\pi\)
−0.530637 + 0.847599i \(0.678047\pi\)
\(864\) −5.19615 −0.176777
\(865\) 0 0
\(866\) 58.8897 2.00115
\(867\) −19.0000 −0.645274
\(868\) −12.0000 −0.407307
\(869\) 27.7128 0.940093
\(870\) 0 0
\(871\) 0 0
\(872\) −12.0000 −0.406371
\(873\) 13.8564 0.468968
\(874\) 0 0
\(875\) 0 0
\(876\) 0 0
\(877\) −48.4974 −1.63764 −0.818821 0.574049i \(-0.805372\pi\)
−0.818821 + 0.574049i \(0.805372\pi\)
\(878\) 13.8564 0.467631
\(879\) −27.7128 −0.934730
\(880\) 0 0
\(881\) −18.0000 −0.606435 −0.303218 0.952921i \(-0.598061\pi\)
−0.303218 + 0.952921i \(0.598061\pi\)
\(882\) −8.66025 −0.291606
\(883\) 20.0000 0.673054 0.336527 0.941674i \(-0.390748\pi\)
0.336527 + 0.941674i \(0.390748\pi\)
\(884\) 0 0
\(885\) 0 0
\(886\) 62.3538 2.09482
\(887\) 48.0000 1.61168 0.805841 0.592132i \(-0.201714\pi\)
0.805841 + 0.592132i \(0.201714\pi\)
\(888\) −12.0000 −0.402694
\(889\) −27.7128 −0.929458
\(890\) 0 0
\(891\) −3.46410 −0.116052
\(892\) 3.46410 0.115987
\(893\) −12.0000 −0.401565
\(894\) −24.0000 −0.802680
\(895\) 0 0
\(896\) 42.0000 1.40312
\(897\) 0 0
\(898\) −12.0000 −0.400445
\(899\) 20.7846 0.693206
\(900\) −5.00000 −0.166667
\(901\) 36.0000 1.19933
\(902\) 41.5692 1.38410
\(903\) 13.8564 0.461112
\(904\) −10.3923 −0.345643
\(905\) 0 0
\(906\) 18.0000 0.598010
\(907\) 44.0000 1.46100 0.730498 0.682915i \(-0.239288\pi\)
0.730498 + 0.682915i \(0.239288\pi\)
\(908\) 17.3205 0.574801
\(909\) −6.00000 −0.199007
\(910\) 0 0
\(911\) 24.0000 0.795155 0.397578 0.917568i \(-0.369851\pi\)
0.397578 + 0.917568i \(0.369851\pi\)
\(912\) −17.3205 −0.573539
\(913\) −12.0000 −0.397142
\(914\) −48.0000 −1.58770
\(915\) 0 0
\(916\) −6.92820 −0.228914
\(917\) 41.5692 1.37274
\(918\) 10.3923 0.342997
\(919\) −32.0000 −1.05558 −0.527791 0.849374i \(-0.676980\pi\)
−0.527791 + 0.849374i \(0.676980\pi\)
\(920\) 0 0
\(921\) 10.3923 0.342438
\(922\) −24.0000 −0.790398
\(923\) 0 0
\(924\) −12.0000 −0.394771
\(925\) −34.6410 −1.13899
\(926\) 30.0000 0.985861
\(927\) −8.00000 −0.262754
\(928\) 31.1769 1.02343
\(929\) 20.7846 0.681921 0.340960 0.940078i \(-0.389248\pi\)
0.340960 + 0.940078i \(0.389248\pi\)
\(930\) 0 0
\(931\) −17.3205 −0.567657
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 20.7846 0.680093
\(935\) 0 0
\(936\) 0 0
\(937\) −38.0000 −1.24141 −0.620703 0.784046i \(-0.713153\pi\)
−0.620703 + 0.784046i \(0.713153\pi\)
\(938\) 62.3538 2.03592
\(939\) −10.0000 −0.326338
\(940\) 0 0
\(941\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(942\) 24.2487 0.790066
\(943\) 0 0
\(944\) −51.9615 −1.69120
\(945\) 0 0
\(946\) 24.0000 0.780307
\(947\) 51.9615 1.68852 0.844261 0.535932i \(-0.180040\pi\)
0.844261 + 0.535932i \(0.180040\pi\)
\(948\) 8.00000 0.259828
\(949\) 0 0
\(950\) −30.0000 −0.973329
\(951\) 13.8564 0.449325
\(952\) −36.0000 −1.16677
\(953\) −42.0000 −1.36051 −0.680257 0.732974i \(-0.738132\pi\)
−0.680257 + 0.732974i \(0.738132\pi\)
\(954\) −10.3923 −0.336463
\(955\) 0 0
\(956\) 10.3923 0.336111
\(957\) 20.7846 0.671871
\(958\) 18.0000 0.581554
\(959\) 72.0000 2.32500
\(960\) 0 0
\(961\) −19.0000 −0.612903
\(962\) 0 0
\(963\) 12.0000 0.386695
\(964\) 13.8564 0.446285
\(965\) 0 0
\(966\) 0 0
\(967\) −10.3923 −0.334194 −0.167097 0.985940i \(-0.553439\pi\)
−0.167097 + 0.985940i \(0.553439\pi\)
\(968\) 1.73205 0.0556702
\(969\) 20.7846 0.667698
\(970\) 0 0
\(971\) −12.0000 −0.385098 −0.192549 0.981287i \(-0.561675\pi\)
−0.192549 + 0.981287i \(0.561675\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 13.8564 0.444216
\(974\) 66.0000 2.11478
\(975\) 0 0
\(976\) 10.0000 0.320092
\(977\) −48.4974 −1.55157 −0.775785 0.630997i \(-0.782646\pi\)
−0.775785 + 0.630997i \(0.782646\pi\)
\(978\) 6.00000 0.191859
\(979\) 24.0000 0.767043
\(980\) 0 0
\(981\) −6.92820 −0.221201
\(982\) −20.7846 −0.663264
\(983\) −51.9615 −1.65732 −0.828658 0.559756i \(-0.810895\pi\)
−0.828658 + 0.559756i \(0.810895\pi\)
\(984\) −12.0000 −0.382546
\(985\) 0 0
\(986\) −62.3538 −1.98575
\(987\) 12.0000 0.381964
\(988\) 0 0
\(989\) 0 0
\(990\) 0 0
\(991\) 32.0000 1.01651 0.508257 0.861206i \(-0.330290\pi\)
0.508257 + 0.861206i \(0.330290\pi\)
\(992\) 18.0000 0.571501
\(993\) 3.46410 0.109930
\(994\) −20.7846 −0.659248
\(995\) 0 0
\(996\) −3.46410 −0.109764
\(997\) 38.0000 1.20347 0.601736 0.798695i \(-0.294476\pi\)
0.601736 + 0.798695i \(0.294476\pi\)
\(998\) −18.0000 −0.569780
\(999\) −6.92820 −0.219199
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 507.2.a.f.1.1 2
3.2 odd 2 1521.2.a.l.1.2 2
4.3 odd 2 8112.2.a.bv.1.2 2
13.2 odd 12 507.2.j.a.316.1 2
13.3 even 3 507.2.e.e.22.2 4
13.4 even 6 507.2.e.e.484.1 4
13.5 odd 4 39.2.b.a.25.2 yes 2
13.6 odd 12 507.2.j.c.361.1 2
13.7 odd 12 507.2.j.a.361.1 2
13.8 odd 4 39.2.b.a.25.1 2
13.9 even 3 507.2.e.e.484.2 4
13.10 even 6 507.2.e.e.22.1 4
13.11 odd 12 507.2.j.c.316.1 2
13.12 even 2 inner 507.2.a.f.1.2 2
39.5 even 4 117.2.b.a.64.1 2
39.8 even 4 117.2.b.a.64.2 2
39.38 odd 2 1521.2.a.l.1.1 2
52.31 even 4 624.2.c.e.337.2 2
52.47 even 4 624.2.c.e.337.1 2
52.51 odd 2 8112.2.a.bv.1.1 2
65.8 even 4 975.2.h.f.649.1 4
65.18 even 4 975.2.h.f.649.3 4
65.34 odd 4 975.2.b.d.376.2 2
65.44 odd 4 975.2.b.d.376.1 2
65.47 even 4 975.2.h.f.649.4 4
65.57 even 4 975.2.h.f.649.2 4
91.34 even 4 1911.2.c.d.883.1 2
91.83 even 4 1911.2.c.d.883.2 2
104.5 odd 4 2496.2.c.k.961.1 2
104.21 odd 4 2496.2.c.k.961.2 2
104.83 even 4 2496.2.c.d.961.2 2
104.99 even 4 2496.2.c.d.961.1 2
156.47 odd 4 1872.2.c.e.1585.1 2
156.83 odd 4 1872.2.c.e.1585.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
39.2.b.a.25.1 2 13.8 odd 4
39.2.b.a.25.2 yes 2 13.5 odd 4
117.2.b.a.64.1 2 39.5 even 4
117.2.b.a.64.2 2 39.8 even 4
507.2.a.f.1.1 2 1.1 even 1 trivial
507.2.a.f.1.2 2 13.12 even 2 inner
507.2.e.e.22.1 4 13.10 even 6
507.2.e.e.22.2 4 13.3 even 3
507.2.e.e.484.1 4 13.4 even 6
507.2.e.e.484.2 4 13.9 even 3
507.2.j.a.316.1 2 13.2 odd 12
507.2.j.a.361.1 2 13.7 odd 12
507.2.j.c.316.1 2 13.11 odd 12
507.2.j.c.361.1 2 13.6 odd 12
624.2.c.e.337.1 2 52.47 even 4
624.2.c.e.337.2 2 52.31 even 4
975.2.b.d.376.1 2 65.44 odd 4
975.2.b.d.376.2 2 65.34 odd 4
975.2.h.f.649.1 4 65.8 even 4
975.2.h.f.649.2 4 65.57 even 4
975.2.h.f.649.3 4 65.18 even 4
975.2.h.f.649.4 4 65.47 even 4
1521.2.a.l.1.1 2 39.38 odd 2
1521.2.a.l.1.2 2 3.2 odd 2
1872.2.c.e.1585.1 2 156.47 odd 4
1872.2.c.e.1585.2 2 156.83 odd 4
1911.2.c.d.883.1 2 91.34 even 4
1911.2.c.d.883.2 2 91.83 even 4
2496.2.c.d.961.1 2 104.99 even 4
2496.2.c.d.961.2 2 104.83 even 4
2496.2.c.k.961.1 2 104.5 odd 4
2496.2.c.k.961.2 2 104.21 odd 4
8112.2.a.bv.1.1 2 52.51 odd 2
8112.2.a.bv.1.2 2 4.3 odd 2