Properties

Label 4650.2.a.by.1.2
Level $4650$
Weight $2$
Character 4650.1
Self dual yes
Analytic conductor $37.130$
Analytic rank $1$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 4650 = 2 \cdot 3 \cdot 5^{2} \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 4650.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(37.1304369399\)
Analytic rank: \(1\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \( x^{2} - x - 8 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 930)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 4650.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{6} +2.37228 q^{7} -1.00000 q^{8} +1.00000 q^{9} +6.37228 q^{11} -1.00000 q^{12} +2.00000 q^{13} -2.37228 q^{14} +1.00000 q^{16} -6.74456 q^{17} -1.00000 q^{18} -6.37228 q^{19} -2.37228 q^{21} -6.37228 q^{22} +2.37228 q^{23} +1.00000 q^{24} -2.00000 q^{26} -1.00000 q^{27} +2.37228 q^{28} +2.74456 q^{29} -1.00000 q^{31} -1.00000 q^{32} -6.37228 q^{33} +6.74456 q^{34} +1.00000 q^{36} -10.7446 q^{37} +6.37228 q^{38} -2.00000 q^{39} -10.7446 q^{41} +2.37228 q^{42} -6.37228 q^{43} +6.37228 q^{44} -2.37228 q^{46} -4.74456 q^{47} -1.00000 q^{48} -1.37228 q^{49} +6.74456 q^{51} +2.00000 q^{52} +4.37228 q^{53} +1.00000 q^{54} -2.37228 q^{56} +6.37228 q^{57} -2.74456 q^{58} -8.74456 q^{59} -11.4891 q^{61} +1.00000 q^{62} +2.37228 q^{63} +1.00000 q^{64} +6.37228 q^{66} +0.744563 q^{67} -6.74456 q^{68} -2.37228 q^{69} -2.37228 q^{71} -1.00000 q^{72} -9.11684 q^{73} +10.7446 q^{74} -6.37228 q^{76} +15.1168 q^{77} +2.00000 q^{78} -10.3723 q^{79} +1.00000 q^{81} +10.7446 q^{82} +12.0000 q^{83} -2.37228 q^{84} +6.37228 q^{86} -2.74456 q^{87} -6.37228 q^{88} +4.37228 q^{89} +4.74456 q^{91} +2.37228 q^{92} +1.00000 q^{93} +4.74456 q^{94} +1.00000 q^{96} -2.00000 q^{97} +1.37228 q^{98} +6.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q - 2 q^{2} - 2 q^{3} + 2 q^{4} + 2 q^{6} - q^{7} - 2 q^{8} + 2 q^{9} + 7 q^{11} - 2 q^{12} + 4 q^{13} + q^{14} + 2 q^{16} - 2 q^{17} - 2 q^{18} - 7 q^{19} + q^{21} - 7 q^{22} - q^{23} + 2 q^{24} - 4 q^{26} - 2 q^{27} - q^{28} - 6 q^{29} - 2 q^{31} - 2 q^{32} - 7 q^{33} + 2 q^{34} + 2 q^{36} - 10 q^{37} + 7 q^{38} - 4 q^{39} - 10 q^{41} - q^{42} - 7 q^{43} + 7 q^{44} + q^{46} + 2 q^{47} - 2 q^{48} + 3 q^{49} + 2 q^{51} + 4 q^{52} + 3 q^{53} + 2 q^{54} + q^{56} + 7 q^{57} + 6 q^{58} - 6 q^{59} + 2 q^{62} - q^{63} + 2 q^{64} + 7 q^{66} - 10 q^{67} - 2 q^{68} + q^{69} + q^{71} - 2 q^{72} - q^{73} + 10 q^{74} - 7 q^{76} + 13 q^{77} + 4 q^{78} - 15 q^{79} + 2 q^{81} + 10 q^{82} + 24 q^{83} + q^{84} + 7 q^{86} + 6 q^{87} - 7 q^{88} + 3 q^{89} - 2 q^{91} - q^{92} + 2 q^{93} - 2 q^{94} + 2 q^{96} - 4 q^{97} - 3 q^{98} + 7 q^{99}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) 1.00000 0.408248
\(7\) 2.37228 0.896638 0.448319 0.893874i \(-0.352023\pi\)
0.448319 + 0.893874i \(0.352023\pi\)
\(8\) −1.00000 −0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) 6.37228 1.92132 0.960658 0.277736i \(-0.0895839\pi\)
0.960658 + 0.277736i \(0.0895839\pi\)
\(12\) −1.00000 −0.288675
\(13\) 2.00000 0.554700 0.277350 0.960769i \(-0.410544\pi\)
0.277350 + 0.960769i \(0.410544\pi\)
\(14\) −2.37228 −0.634019
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −6.74456 −1.63580 −0.817898 0.575363i \(-0.804861\pi\)
−0.817898 + 0.575363i \(0.804861\pi\)
\(18\) −1.00000 −0.235702
\(19\) −6.37228 −1.46190 −0.730951 0.682430i \(-0.760923\pi\)
−0.730951 + 0.682430i \(0.760923\pi\)
\(20\) 0 0
\(21\) −2.37228 −0.517674
\(22\) −6.37228 −1.35857
\(23\) 2.37228 0.494655 0.247327 0.968932i \(-0.420448\pi\)
0.247327 + 0.968932i \(0.420448\pi\)
\(24\) 1.00000 0.204124
\(25\) 0 0
\(26\) −2.00000 −0.392232
\(27\) −1.00000 −0.192450
\(28\) 2.37228 0.448319
\(29\) 2.74456 0.509652 0.254826 0.966987i \(-0.417982\pi\)
0.254826 + 0.966987i \(0.417982\pi\)
\(30\) 0 0
\(31\) −1.00000 −0.179605
\(32\) −1.00000 −0.176777
\(33\) −6.37228 −1.10927
\(34\) 6.74456 1.15668
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −10.7446 −1.76640 −0.883198 0.469001i \(-0.844614\pi\)
−0.883198 + 0.469001i \(0.844614\pi\)
\(38\) 6.37228 1.03372
\(39\) −2.00000 −0.320256
\(40\) 0 0
\(41\) −10.7446 −1.67802 −0.839009 0.544117i \(-0.816865\pi\)
−0.839009 + 0.544117i \(0.816865\pi\)
\(42\) 2.37228 0.366051
\(43\) −6.37228 −0.971764 −0.485882 0.874024i \(-0.661501\pi\)
−0.485882 + 0.874024i \(0.661501\pi\)
\(44\) 6.37228 0.960658
\(45\) 0 0
\(46\) −2.37228 −0.349774
\(47\) −4.74456 −0.692066 −0.346033 0.938222i \(-0.612471\pi\)
−0.346033 + 0.938222i \(0.612471\pi\)
\(48\) −1.00000 −0.144338
\(49\) −1.37228 −0.196040
\(50\) 0 0
\(51\) 6.74456 0.944428
\(52\) 2.00000 0.277350
\(53\) 4.37228 0.600579 0.300290 0.953848i \(-0.402917\pi\)
0.300290 + 0.953848i \(0.402917\pi\)
\(54\) 1.00000 0.136083
\(55\) 0 0
\(56\) −2.37228 −0.317009
\(57\) 6.37228 0.844029
\(58\) −2.74456 −0.360379
\(59\) −8.74456 −1.13845 −0.569223 0.822183i \(-0.692756\pi\)
−0.569223 + 0.822183i \(0.692756\pi\)
\(60\) 0 0
\(61\) −11.4891 −1.47103 −0.735516 0.677507i \(-0.763060\pi\)
−0.735516 + 0.677507i \(0.763060\pi\)
\(62\) 1.00000 0.127000
\(63\) 2.37228 0.298879
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 6.37228 0.784374
\(67\) 0.744563 0.0909628 0.0454814 0.998965i \(-0.485518\pi\)
0.0454814 + 0.998965i \(0.485518\pi\)
\(68\) −6.74456 −0.817898
\(69\) −2.37228 −0.285589
\(70\) 0 0
\(71\) −2.37228 −0.281538 −0.140769 0.990042i \(-0.544957\pi\)
−0.140769 + 0.990042i \(0.544957\pi\)
\(72\) −1.00000 −0.117851
\(73\) −9.11684 −1.06705 −0.533523 0.845786i \(-0.679132\pi\)
−0.533523 + 0.845786i \(0.679132\pi\)
\(74\) 10.7446 1.24903
\(75\) 0 0
\(76\) −6.37228 −0.730951
\(77\) 15.1168 1.72272
\(78\) 2.00000 0.226455
\(79\) −10.3723 −1.16697 −0.583486 0.812123i \(-0.698312\pi\)
−0.583486 + 0.812123i \(0.698312\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 10.7446 1.18654
\(83\) 12.0000 1.31717 0.658586 0.752506i \(-0.271155\pi\)
0.658586 + 0.752506i \(0.271155\pi\)
\(84\) −2.37228 −0.258837
\(85\) 0 0
\(86\) 6.37228 0.687141
\(87\) −2.74456 −0.294248
\(88\) −6.37228 −0.679287
\(89\) 4.37228 0.463461 0.231730 0.972780i \(-0.425561\pi\)
0.231730 + 0.972780i \(0.425561\pi\)
\(90\) 0 0
\(91\) 4.74456 0.497365
\(92\) 2.37228 0.247327
\(93\) 1.00000 0.103695
\(94\) 4.74456 0.489364
\(95\) 0 0
\(96\) 1.00000 0.102062
\(97\) −2.00000 −0.203069 −0.101535 0.994832i \(-0.532375\pi\)
−0.101535 + 0.994832i \(0.532375\pi\)
\(98\) 1.37228 0.138621
\(99\) 6.37228 0.640438
\(100\) 0 0
\(101\) −9.11684 −0.907160 −0.453580 0.891216i \(-0.649853\pi\)
−0.453580 + 0.891216i \(0.649853\pi\)
\(102\) −6.74456 −0.667811
\(103\) 8.00000 0.788263 0.394132 0.919054i \(-0.371045\pi\)
0.394132 + 0.919054i \(0.371045\pi\)
\(104\) −2.00000 −0.196116
\(105\) 0 0
\(106\) −4.37228 −0.424674
\(107\) −6.37228 −0.616032 −0.308016 0.951381i \(-0.599665\pi\)
−0.308016 + 0.951381i \(0.599665\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −6.74456 −0.646012 −0.323006 0.946397i \(-0.604693\pi\)
−0.323006 + 0.946397i \(0.604693\pi\)
\(110\) 0 0
\(111\) 10.7446 1.01983
\(112\) 2.37228 0.224160
\(113\) 8.37228 0.787598 0.393799 0.919197i \(-0.371161\pi\)
0.393799 + 0.919197i \(0.371161\pi\)
\(114\) −6.37228 −0.596819
\(115\) 0 0
\(116\) 2.74456 0.254826
\(117\) 2.00000 0.184900
\(118\) 8.74456 0.805002
\(119\) −16.0000 −1.46672
\(120\) 0 0
\(121\) 29.6060 2.69145
\(122\) 11.4891 1.04018
\(123\) 10.7446 0.968805
\(124\) −1.00000 −0.0898027
\(125\) 0 0
\(126\) −2.37228 −0.211340
\(127\) 9.48913 0.842024 0.421012 0.907055i \(-0.361675\pi\)
0.421012 + 0.907055i \(0.361675\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 6.37228 0.561048
\(130\) 0 0
\(131\) 18.2337 1.59308 0.796542 0.604583i \(-0.206660\pi\)
0.796542 + 0.604583i \(0.206660\pi\)
\(132\) −6.37228 −0.554636
\(133\) −15.1168 −1.31080
\(134\) −0.744563 −0.0643204
\(135\) 0 0
\(136\) 6.74456 0.578341
\(137\) −19.4891 −1.66507 −0.832534 0.553974i \(-0.813111\pi\)
−0.832534 + 0.553974i \(0.813111\pi\)
\(138\) 2.37228 0.201942
\(139\) 0.744563 0.0631530 0.0315765 0.999501i \(-0.489947\pi\)
0.0315765 + 0.999501i \(0.489947\pi\)
\(140\) 0 0
\(141\) 4.74456 0.399564
\(142\) 2.37228 0.199077
\(143\) 12.7446 1.06575
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) 9.11684 0.754515
\(147\) 1.37228 0.113184
\(148\) −10.7446 −0.883198
\(149\) 5.11684 0.419188 0.209594 0.977788i \(-0.432786\pi\)
0.209594 + 0.977788i \(0.432786\pi\)
\(150\) 0 0
\(151\) −8.00000 −0.651031 −0.325515 0.945537i \(-0.605538\pi\)
−0.325515 + 0.945537i \(0.605538\pi\)
\(152\) 6.37228 0.516860
\(153\) −6.74456 −0.545266
\(154\) −15.1168 −1.21815
\(155\) 0 0
\(156\) −2.00000 −0.160128
\(157\) −3.62772 −0.289523 −0.144762 0.989467i \(-0.546242\pi\)
−0.144762 + 0.989467i \(0.546242\pi\)
\(158\) 10.3723 0.825174
\(159\) −4.37228 −0.346744
\(160\) 0 0
\(161\) 5.62772 0.443526
\(162\) −1.00000 −0.0785674
\(163\) 10.2337 0.801564 0.400782 0.916173i \(-0.368738\pi\)
0.400782 + 0.916173i \(0.368738\pi\)
\(164\) −10.7446 −0.839009
\(165\) 0 0
\(166\) −12.0000 −0.931381
\(167\) 18.3723 1.42169 0.710845 0.703349i \(-0.248313\pi\)
0.710845 + 0.703349i \(0.248313\pi\)
\(168\) 2.37228 0.183025
\(169\) −9.00000 −0.692308
\(170\) 0 0
\(171\) −6.37228 −0.487301
\(172\) −6.37228 −0.485882
\(173\) 18.0000 1.36851 0.684257 0.729241i \(-0.260127\pi\)
0.684257 + 0.729241i \(0.260127\pi\)
\(174\) 2.74456 0.208065
\(175\) 0 0
\(176\) 6.37228 0.480329
\(177\) 8.74456 0.657282
\(178\) −4.37228 −0.327716
\(179\) −12.0000 −0.896922 −0.448461 0.893802i \(-0.648028\pi\)
−0.448461 + 0.893802i \(0.648028\pi\)
\(180\) 0 0
\(181\) −13.8614 −1.03031 −0.515155 0.857097i \(-0.672266\pi\)
−0.515155 + 0.857097i \(0.672266\pi\)
\(182\) −4.74456 −0.351690
\(183\) 11.4891 0.849301
\(184\) −2.37228 −0.174887
\(185\) 0 0
\(186\) −1.00000 −0.0733236
\(187\) −42.9783 −3.14288
\(188\) −4.74456 −0.346033
\(189\) −2.37228 −0.172558
\(190\) 0 0
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) −1.00000 −0.0721688
\(193\) 7.48913 0.539079 0.269540 0.962989i \(-0.413129\pi\)
0.269540 + 0.962989i \(0.413129\pi\)
\(194\) 2.00000 0.143592
\(195\) 0 0
\(196\) −1.37228 −0.0980201
\(197\) 3.48913 0.248590 0.124295 0.992245i \(-0.460333\pi\)
0.124295 + 0.992245i \(0.460333\pi\)
\(198\) −6.37228 −0.452858
\(199\) −18.3723 −1.30238 −0.651188 0.758916i \(-0.725729\pi\)
−0.651188 + 0.758916i \(0.725729\pi\)
\(200\) 0 0
\(201\) −0.744563 −0.0525174
\(202\) 9.11684 0.641459
\(203\) 6.51087 0.456974
\(204\) 6.74456 0.472214
\(205\) 0 0
\(206\) −8.00000 −0.557386
\(207\) 2.37228 0.164885
\(208\) 2.00000 0.138675
\(209\) −40.6060 −2.80877
\(210\) 0 0
\(211\) −6.37228 −0.438686 −0.219343 0.975648i \(-0.570391\pi\)
−0.219343 + 0.975648i \(0.570391\pi\)
\(212\) 4.37228 0.300290
\(213\) 2.37228 0.162546
\(214\) 6.37228 0.435600
\(215\) 0 0
\(216\) 1.00000 0.0680414
\(217\) −2.37228 −0.161041
\(218\) 6.74456 0.456799
\(219\) 9.11684 0.616059
\(220\) 0 0
\(221\) −13.4891 −0.907377
\(222\) −10.7446 −0.721128
\(223\) −20.7446 −1.38916 −0.694579 0.719416i \(-0.744409\pi\)
−0.694579 + 0.719416i \(0.744409\pi\)
\(224\) −2.37228 −0.158505
\(225\) 0 0
\(226\) −8.37228 −0.556916
\(227\) 11.1168 0.737851 0.368925 0.929459i \(-0.379726\pi\)
0.368925 + 0.929459i \(0.379726\pi\)
\(228\) 6.37228 0.422015
\(229\) 21.1168 1.39544 0.697720 0.716370i \(-0.254198\pi\)
0.697720 + 0.716370i \(0.254198\pi\)
\(230\) 0 0
\(231\) −15.1168 −0.994615
\(232\) −2.74456 −0.180189
\(233\) −13.8614 −0.908091 −0.454045 0.890979i \(-0.650020\pi\)
−0.454045 + 0.890979i \(0.650020\pi\)
\(234\) −2.00000 −0.130744
\(235\) 0 0
\(236\) −8.74456 −0.569223
\(237\) 10.3723 0.673752
\(238\) 16.0000 1.03713
\(239\) −6.51087 −0.421153 −0.210577 0.977577i \(-0.567534\pi\)
−0.210577 + 0.977577i \(0.567534\pi\)
\(240\) 0 0
\(241\) 27.4891 1.77073 0.885365 0.464896i \(-0.153908\pi\)
0.885365 + 0.464896i \(0.153908\pi\)
\(242\) −29.6060 −1.90314
\(243\) −1.00000 −0.0641500
\(244\) −11.4891 −0.735516
\(245\) 0 0
\(246\) −10.7446 −0.685048
\(247\) −12.7446 −0.810917
\(248\) 1.00000 0.0635001
\(249\) −12.0000 −0.760469
\(250\) 0 0
\(251\) −4.00000 −0.252478 −0.126239 0.992000i \(-0.540291\pi\)
−0.126239 + 0.992000i \(0.540291\pi\)
\(252\) 2.37228 0.149440
\(253\) 15.1168 0.950388
\(254\) −9.48913 −0.595401
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −7.62772 −0.475804 −0.237902 0.971289i \(-0.576460\pi\)
−0.237902 + 0.971289i \(0.576460\pi\)
\(258\) −6.37228 −0.396721
\(259\) −25.4891 −1.58382
\(260\) 0 0
\(261\) 2.74456 0.169884
\(262\) −18.2337 −1.12648
\(263\) −26.9783 −1.66355 −0.831775 0.555113i \(-0.812675\pi\)
−0.831775 + 0.555113i \(0.812675\pi\)
\(264\) 6.37228 0.392187
\(265\) 0 0
\(266\) 15.1168 0.926873
\(267\) −4.37228 −0.267579
\(268\) 0.744563 0.0454814
\(269\) −2.00000 −0.121942 −0.0609711 0.998140i \(-0.519420\pi\)
−0.0609711 + 0.998140i \(0.519420\pi\)
\(270\) 0 0
\(271\) −31.1168 −1.89021 −0.945107 0.326762i \(-0.894043\pi\)
−0.945107 + 0.326762i \(0.894043\pi\)
\(272\) −6.74456 −0.408949
\(273\) −4.74456 −0.287154
\(274\) 19.4891 1.17738
\(275\) 0 0
\(276\) −2.37228 −0.142795
\(277\) −26.7446 −1.60693 −0.803463 0.595355i \(-0.797011\pi\)
−0.803463 + 0.595355i \(0.797011\pi\)
\(278\) −0.744563 −0.0446559
\(279\) −1.00000 −0.0598684
\(280\) 0 0
\(281\) 5.25544 0.313513 0.156757 0.987637i \(-0.449896\pi\)
0.156757 + 0.987637i \(0.449896\pi\)
\(282\) −4.74456 −0.282535
\(283\) −12.0000 −0.713326 −0.356663 0.934233i \(-0.616086\pi\)
−0.356663 + 0.934233i \(0.616086\pi\)
\(284\) −2.37228 −0.140769
\(285\) 0 0
\(286\) −12.7446 −0.753602
\(287\) −25.4891 −1.50458
\(288\) −1.00000 −0.0589256
\(289\) 28.4891 1.67583
\(290\) 0 0
\(291\) 2.00000 0.117242
\(292\) −9.11684 −0.533523
\(293\) −15.4891 −0.904884 −0.452442 0.891794i \(-0.649447\pi\)
−0.452442 + 0.891794i \(0.649447\pi\)
\(294\) −1.37228 −0.0800331
\(295\) 0 0
\(296\) 10.7446 0.624515
\(297\) −6.37228 −0.369757
\(298\) −5.11684 −0.296411
\(299\) 4.74456 0.274385
\(300\) 0 0
\(301\) −15.1168 −0.871320
\(302\) 8.00000 0.460348
\(303\) 9.11684 0.523749
\(304\) −6.37228 −0.365475
\(305\) 0 0
\(306\) 6.74456 0.385561
\(307\) 14.9783 0.854854 0.427427 0.904050i \(-0.359420\pi\)
0.427427 + 0.904050i \(0.359420\pi\)
\(308\) 15.1168 0.861362
\(309\) −8.00000 −0.455104
\(310\) 0 0
\(311\) 8.00000 0.453638 0.226819 0.973937i \(-0.427167\pi\)
0.226819 + 0.973937i \(0.427167\pi\)
\(312\) 2.00000 0.113228
\(313\) −10.0000 −0.565233 −0.282617 0.959233i \(-0.591202\pi\)
−0.282617 + 0.959233i \(0.591202\pi\)
\(314\) 3.62772 0.204724
\(315\) 0 0
\(316\) −10.3723 −0.583486
\(317\) 13.2554 0.744500 0.372250 0.928133i \(-0.378586\pi\)
0.372250 + 0.928133i \(0.378586\pi\)
\(318\) 4.37228 0.245185
\(319\) 17.4891 0.979203
\(320\) 0 0
\(321\) 6.37228 0.355666
\(322\) −5.62772 −0.313621
\(323\) 42.9783 2.39137
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) −10.2337 −0.566792
\(327\) 6.74456 0.372975
\(328\) 10.7446 0.593269
\(329\) −11.2554 −0.620532
\(330\) 0 0
\(331\) 21.4891 1.18115 0.590575 0.806983i \(-0.298901\pi\)
0.590575 + 0.806983i \(0.298901\pi\)
\(332\) 12.0000 0.658586
\(333\) −10.7446 −0.588798
\(334\) −18.3723 −1.00529
\(335\) 0 0
\(336\) −2.37228 −0.129419
\(337\) 16.9783 0.924864 0.462432 0.886655i \(-0.346977\pi\)
0.462432 + 0.886655i \(0.346977\pi\)
\(338\) 9.00000 0.489535
\(339\) −8.37228 −0.454720
\(340\) 0 0
\(341\) −6.37228 −0.345078
\(342\) 6.37228 0.344574
\(343\) −19.8614 −1.07242
\(344\) 6.37228 0.343570
\(345\) 0 0
\(346\) −18.0000 −0.967686
\(347\) 32.4674 1.74294 0.871470 0.490449i \(-0.163167\pi\)
0.871470 + 0.490449i \(0.163167\pi\)
\(348\) −2.74456 −0.147124
\(349\) 18.7446 1.00337 0.501687 0.865049i \(-0.332713\pi\)
0.501687 + 0.865049i \(0.332713\pi\)
\(350\) 0 0
\(351\) −2.00000 −0.106752
\(352\) −6.37228 −0.339644
\(353\) 7.48913 0.398606 0.199303 0.979938i \(-0.436132\pi\)
0.199303 + 0.979938i \(0.436132\pi\)
\(354\) −8.74456 −0.464768
\(355\) 0 0
\(356\) 4.37228 0.231730
\(357\) 16.0000 0.846810
\(358\) 12.0000 0.634220
\(359\) −2.37228 −0.125204 −0.0626021 0.998039i \(-0.519940\pi\)
−0.0626021 + 0.998039i \(0.519940\pi\)
\(360\) 0 0
\(361\) 21.6060 1.13716
\(362\) 13.8614 0.728539
\(363\) −29.6060 −1.55391
\(364\) 4.74456 0.248683
\(365\) 0 0
\(366\) −11.4891 −0.600546
\(367\) −14.2337 −0.742992 −0.371496 0.928434i \(-0.621155\pi\)
−0.371496 + 0.928434i \(0.621155\pi\)
\(368\) 2.37228 0.123664
\(369\) −10.7446 −0.559340
\(370\) 0 0
\(371\) 10.3723 0.538502
\(372\) 1.00000 0.0518476
\(373\) 13.8614 0.717716 0.358858 0.933392i \(-0.383166\pi\)
0.358858 + 0.933392i \(0.383166\pi\)
\(374\) 42.9783 2.22235
\(375\) 0 0
\(376\) 4.74456 0.244682
\(377\) 5.48913 0.282704
\(378\) 2.37228 0.122017
\(379\) −0.138593 −0.00711906 −0.00355953 0.999994i \(-0.501133\pi\)
−0.00355953 + 0.999994i \(0.501133\pi\)
\(380\) 0 0
\(381\) −9.48913 −0.486143
\(382\) −16.0000 −0.818631
\(383\) 9.48913 0.484872 0.242436 0.970167i \(-0.422054\pi\)
0.242436 + 0.970167i \(0.422054\pi\)
\(384\) 1.00000 0.0510310
\(385\) 0 0
\(386\) −7.48913 −0.381186
\(387\) −6.37228 −0.323921
\(388\) −2.00000 −0.101535
\(389\) −28.9783 −1.46926 −0.734628 0.678470i \(-0.762643\pi\)
−0.734628 + 0.678470i \(0.762643\pi\)
\(390\) 0 0
\(391\) −16.0000 −0.809155
\(392\) 1.37228 0.0693107
\(393\) −18.2337 −0.919768
\(394\) −3.48913 −0.175780
\(395\) 0 0
\(396\) 6.37228 0.320219
\(397\) −38.6060 −1.93758 −0.968789 0.247887i \(-0.920264\pi\)
−0.968789 + 0.247887i \(0.920264\pi\)
\(398\) 18.3723 0.920919
\(399\) 15.1168 0.756789
\(400\) 0 0
\(401\) 28.3723 1.41684 0.708422 0.705789i \(-0.249407\pi\)
0.708422 + 0.705789i \(0.249407\pi\)
\(402\) 0.744563 0.0371354
\(403\) −2.00000 −0.0996271
\(404\) −9.11684 −0.453580
\(405\) 0 0
\(406\) −6.51087 −0.323129
\(407\) −68.4674 −3.39380
\(408\) −6.74456 −0.333906
\(409\) 24.2337 1.19828 0.599139 0.800645i \(-0.295510\pi\)
0.599139 + 0.800645i \(0.295510\pi\)
\(410\) 0 0
\(411\) 19.4891 0.961328
\(412\) 8.00000 0.394132
\(413\) −20.7446 −1.02077
\(414\) −2.37228 −0.116591
\(415\) 0 0
\(416\) −2.00000 −0.0980581
\(417\) −0.744563 −0.0364614
\(418\) 40.6060 1.98610
\(419\) −12.0000 −0.586238 −0.293119 0.956076i \(-0.594693\pi\)
−0.293119 + 0.956076i \(0.594693\pi\)
\(420\) 0 0
\(421\) −3.48913 −0.170050 −0.0850248 0.996379i \(-0.527097\pi\)
−0.0850248 + 0.996379i \(0.527097\pi\)
\(422\) 6.37228 0.310198
\(423\) −4.74456 −0.230689
\(424\) −4.37228 −0.212337
\(425\) 0 0
\(426\) −2.37228 −0.114937
\(427\) −27.2554 −1.31898
\(428\) −6.37228 −0.308016
\(429\) −12.7446 −0.615313
\(430\) 0 0
\(431\) 18.9783 0.914150 0.457075 0.889428i \(-0.348897\pi\)
0.457075 + 0.889428i \(0.348897\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −18.8832 −0.907467 −0.453733 0.891138i \(-0.649908\pi\)
−0.453733 + 0.891138i \(0.649908\pi\)
\(434\) 2.37228 0.113873
\(435\) 0 0
\(436\) −6.74456 −0.323006
\(437\) −15.1168 −0.723137
\(438\) −9.11684 −0.435620
\(439\) 3.25544 0.155374 0.0776868 0.996978i \(-0.475247\pi\)
0.0776868 + 0.996978i \(0.475247\pi\)
\(440\) 0 0
\(441\) −1.37228 −0.0653467
\(442\) 13.4891 0.641612
\(443\) 17.3505 0.824349 0.412174 0.911105i \(-0.364769\pi\)
0.412174 + 0.911105i \(0.364769\pi\)
\(444\) 10.7446 0.509914
\(445\) 0 0
\(446\) 20.7446 0.982284
\(447\) −5.11684 −0.242018
\(448\) 2.37228 0.112080
\(449\) 36.9783 1.74511 0.872556 0.488515i \(-0.162461\pi\)
0.872556 + 0.488515i \(0.162461\pi\)
\(450\) 0 0
\(451\) −68.4674 −3.22400
\(452\) 8.37228 0.393799
\(453\) 8.00000 0.375873
\(454\) −11.1168 −0.521739
\(455\) 0 0
\(456\) −6.37228 −0.298409
\(457\) 34.4674 1.61232 0.806158 0.591700i \(-0.201543\pi\)
0.806158 + 0.591700i \(0.201543\pi\)
\(458\) −21.1168 −0.986725
\(459\) 6.74456 0.314809
\(460\) 0 0
\(461\) 37.7228 1.75693 0.878463 0.477810i \(-0.158569\pi\)
0.878463 + 0.477810i \(0.158569\pi\)
\(462\) 15.1168 0.703299
\(463\) −25.4891 −1.18458 −0.592290 0.805725i \(-0.701776\pi\)
−0.592290 + 0.805725i \(0.701776\pi\)
\(464\) 2.74456 0.127413
\(465\) 0 0
\(466\) 13.8614 0.642117
\(467\) −29.4891 −1.36459 −0.682297 0.731075i \(-0.739019\pi\)
−0.682297 + 0.731075i \(0.739019\pi\)
\(468\) 2.00000 0.0924500
\(469\) 1.76631 0.0815607
\(470\) 0 0
\(471\) 3.62772 0.167156
\(472\) 8.74456 0.402501
\(473\) −40.6060 −1.86706
\(474\) −10.3723 −0.476415
\(475\) 0 0
\(476\) −16.0000 −0.733359
\(477\) 4.37228 0.200193
\(478\) 6.51087 0.297800
\(479\) −19.8614 −0.907491 −0.453745 0.891131i \(-0.649912\pi\)
−0.453745 + 0.891131i \(0.649912\pi\)
\(480\) 0 0
\(481\) −21.4891 −0.979820
\(482\) −27.4891 −1.25210
\(483\) −5.62772 −0.256070
\(484\) 29.6060 1.34573
\(485\) 0 0
\(486\) 1.00000 0.0453609
\(487\) 14.5109 0.657550 0.328775 0.944408i \(-0.393364\pi\)
0.328775 + 0.944408i \(0.393364\pi\)
\(488\) 11.4891 0.520088
\(489\) −10.2337 −0.462783
\(490\) 0 0
\(491\) −12.6060 −0.568899 −0.284450 0.958691i \(-0.591811\pi\)
−0.284450 + 0.958691i \(0.591811\pi\)
\(492\) 10.7446 0.484402
\(493\) −18.5109 −0.833688
\(494\) 12.7446 0.573405
\(495\) 0 0
\(496\) −1.00000 −0.0449013
\(497\) −5.62772 −0.252438
\(498\) 12.0000 0.537733
\(499\) 10.5109 0.470531 0.235266 0.971931i \(-0.424404\pi\)
0.235266 + 0.971931i \(0.424404\pi\)
\(500\) 0 0
\(501\) −18.3723 −0.820813
\(502\) 4.00000 0.178529
\(503\) 17.4891 0.779802 0.389901 0.920857i \(-0.372509\pi\)
0.389901 + 0.920857i \(0.372509\pi\)
\(504\) −2.37228 −0.105670
\(505\) 0 0
\(506\) −15.1168 −0.672026
\(507\) 9.00000 0.399704
\(508\) 9.48913 0.421012
\(509\) −38.7446 −1.71732 −0.858661 0.512543i \(-0.828703\pi\)
−0.858661 + 0.512543i \(0.828703\pi\)
\(510\) 0 0
\(511\) −21.6277 −0.956754
\(512\) −1.00000 −0.0441942
\(513\) 6.37228 0.281343
\(514\) 7.62772 0.336444
\(515\) 0 0
\(516\) 6.37228 0.280524
\(517\) −30.2337 −1.32968
\(518\) 25.4891 1.11993
\(519\) −18.0000 −0.790112
\(520\) 0 0
\(521\) −8.97825 −0.393344 −0.196672 0.980469i \(-0.563013\pi\)
−0.196672 + 0.980469i \(0.563013\pi\)
\(522\) −2.74456 −0.120126
\(523\) −15.8614 −0.693571 −0.346785 0.937944i \(-0.612727\pi\)
−0.346785 + 0.937944i \(0.612727\pi\)
\(524\) 18.2337 0.796542
\(525\) 0 0
\(526\) 26.9783 1.17631
\(527\) 6.74456 0.293798
\(528\) −6.37228 −0.277318
\(529\) −17.3723 −0.755317
\(530\) 0 0
\(531\) −8.74456 −0.379482
\(532\) −15.1168 −0.655398
\(533\) −21.4891 −0.930797
\(534\) 4.37228 0.189207
\(535\) 0 0
\(536\) −0.744563 −0.0321602
\(537\) 12.0000 0.517838
\(538\) 2.00000 0.0862261
\(539\) −8.74456 −0.376655
\(540\) 0 0
\(541\) 7.48913 0.321983 0.160991 0.986956i \(-0.448531\pi\)
0.160991 + 0.986956i \(0.448531\pi\)
\(542\) 31.1168 1.33658
\(543\) 13.8614 0.594850
\(544\) 6.74456 0.289171
\(545\) 0 0
\(546\) 4.74456 0.203049
\(547\) −8.74456 −0.373890 −0.186945 0.982370i \(-0.559859\pi\)
−0.186945 + 0.982370i \(0.559859\pi\)
\(548\) −19.4891 −0.832534
\(549\) −11.4891 −0.490344
\(550\) 0 0
\(551\) −17.4891 −0.745062
\(552\) 2.37228 0.100971
\(553\) −24.6060 −1.04635
\(554\) 26.7446 1.13627
\(555\) 0 0
\(556\) 0.744563 0.0315765
\(557\) −29.1168 −1.23372 −0.616860 0.787073i \(-0.711596\pi\)
−0.616860 + 0.787073i \(0.711596\pi\)
\(558\) 1.00000 0.0423334
\(559\) −12.7446 −0.539038
\(560\) 0 0
\(561\) 42.9783 1.81454
\(562\) −5.25544 −0.221687
\(563\) −38.9783 −1.64274 −0.821369 0.570398i \(-0.806789\pi\)
−0.821369 + 0.570398i \(0.806789\pi\)
\(564\) 4.74456 0.199782
\(565\) 0 0
\(566\) 12.0000 0.504398
\(567\) 2.37228 0.0996265
\(568\) 2.37228 0.0995387
\(569\) 4.37228 0.183296 0.0916478 0.995791i \(-0.470787\pi\)
0.0916478 + 0.995791i \(0.470787\pi\)
\(570\) 0 0
\(571\) −15.2554 −0.638420 −0.319210 0.947684i \(-0.603418\pi\)
−0.319210 + 0.947684i \(0.603418\pi\)
\(572\) 12.7446 0.532877
\(573\) −16.0000 −0.668410
\(574\) 25.4891 1.06390
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) −32.2337 −1.34191 −0.670953 0.741500i \(-0.734115\pi\)
−0.670953 + 0.741500i \(0.734115\pi\)
\(578\) −28.4891 −1.18499
\(579\) −7.48913 −0.311237
\(580\) 0 0
\(581\) 28.4674 1.18103
\(582\) −2.00000 −0.0829027
\(583\) 27.8614 1.15390
\(584\) 9.11684 0.377258
\(585\) 0 0
\(586\) 15.4891 0.639850
\(587\) −12.0000 −0.495293 −0.247647 0.968850i \(-0.579657\pi\)
−0.247647 + 0.968850i \(0.579657\pi\)
\(588\) 1.37228 0.0565919
\(589\) 6.37228 0.262565
\(590\) 0 0
\(591\) −3.48913 −0.143523
\(592\) −10.7446 −0.441599
\(593\) 0.978251 0.0401719 0.0200860 0.999798i \(-0.493606\pi\)
0.0200860 + 0.999798i \(0.493606\pi\)
\(594\) 6.37228 0.261458
\(595\) 0 0
\(596\) 5.11684 0.209594
\(597\) 18.3723 0.751927
\(598\) −4.74456 −0.194020
\(599\) −7.11684 −0.290786 −0.145393 0.989374i \(-0.546445\pi\)
−0.145393 + 0.989374i \(0.546445\pi\)
\(600\) 0 0
\(601\) 35.4891 1.44763 0.723816 0.689993i \(-0.242387\pi\)
0.723816 + 0.689993i \(0.242387\pi\)
\(602\) 15.1168 0.616117
\(603\) 0.744563 0.0303209
\(604\) −8.00000 −0.325515
\(605\) 0 0
\(606\) −9.11684 −0.370346
\(607\) 10.3723 0.420998 0.210499 0.977594i \(-0.432491\pi\)
0.210499 + 0.977594i \(0.432491\pi\)
\(608\) 6.37228 0.258430
\(609\) −6.51087 −0.263834
\(610\) 0 0
\(611\) −9.48913 −0.383889
\(612\) −6.74456 −0.272633
\(613\) −12.5109 −0.505309 −0.252655 0.967557i \(-0.581304\pi\)
−0.252655 + 0.967557i \(0.581304\pi\)
\(614\) −14.9783 −0.604473
\(615\) 0 0
\(616\) −15.1168 −0.609075
\(617\) −25.1168 −1.01117 −0.505583 0.862778i \(-0.668723\pi\)
−0.505583 + 0.862778i \(0.668723\pi\)
\(618\) 8.00000 0.321807
\(619\) −18.2337 −0.732874 −0.366437 0.930443i \(-0.619422\pi\)
−0.366437 + 0.930443i \(0.619422\pi\)
\(620\) 0 0
\(621\) −2.37228 −0.0951964
\(622\) −8.00000 −0.320771
\(623\) 10.3723 0.415557
\(624\) −2.00000 −0.0800641
\(625\) 0 0
\(626\) 10.0000 0.399680
\(627\) 40.6060 1.62165
\(628\) −3.62772 −0.144762
\(629\) 72.4674 2.88946
\(630\) 0 0
\(631\) 5.35053 0.213001 0.106501 0.994313i \(-0.466035\pi\)
0.106501 + 0.994313i \(0.466035\pi\)
\(632\) 10.3723 0.412587
\(633\) 6.37228 0.253275
\(634\) −13.2554 −0.526441
\(635\) 0 0
\(636\) −4.37228 −0.173372
\(637\) −2.74456 −0.108744
\(638\) −17.4891 −0.692401
\(639\) −2.37228 −0.0938460
\(640\) 0 0
\(641\) 18.0000 0.710957 0.355479 0.934684i \(-0.384318\pi\)
0.355479 + 0.934684i \(0.384318\pi\)
\(642\) −6.37228 −0.251494
\(643\) −38.0951 −1.50232 −0.751162 0.660118i \(-0.770506\pi\)
−0.751162 + 0.660118i \(0.770506\pi\)
\(644\) 5.62772 0.221763
\(645\) 0 0
\(646\) −42.9783 −1.69096
\(647\) 35.5842 1.39896 0.699480 0.714652i \(-0.253415\pi\)
0.699480 + 0.714652i \(0.253415\pi\)
\(648\) −1.00000 −0.0392837
\(649\) −55.7228 −2.18731
\(650\) 0 0
\(651\) 2.37228 0.0929770
\(652\) 10.2337 0.400782
\(653\) 4.97825 0.194814 0.0974070 0.995245i \(-0.468945\pi\)
0.0974070 + 0.995245i \(0.468945\pi\)
\(654\) −6.74456 −0.263733
\(655\) 0 0
\(656\) −10.7446 −0.419505
\(657\) −9.11684 −0.355682
\(658\) 11.2554 0.438783
\(659\) 36.0000 1.40236 0.701180 0.712984i \(-0.252657\pi\)
0.701180 + 0.712984i \(0.252657\pi\)
\(660\) 0 0
\(661\) 40.9783 1.59387 0.796935 0.604066i \(-0.206454\pi\)
0.796935 + 0.604066i \(0.206454\pi\)
\(662\) −21.4891 −0.835199
\(663\) 13.4891 0.523874
\(664\) −12.0000 −0.465690
\(665\) 0 0
\(666\) 10.7446 0.416343
\(667\) 6.51087 0.252102
\(668\) 18.3723 0.710845
\(669\) 20.7446 0.802031
\(670\) 0 0
\(671\) −73.2119 −2.82632
\(672\) 2.37228 0.0915127
\(673\) 23.4891 0.905439 0.452720 0.891653i \(-0.350454\pi\)
0.452720 + 0.891653i \(0.350454\pi\)
\(674\) −16.9783 −0.653978
\(675\) 0 0
\(676\) −9.00000 −0.346154
\(677\) 1.39403 0.0535770 0.0267885 0.999641i \(-0.491472\pi\)
0.0267885 + 0.999641i \(0.491472\pi\)
\(678\) 8.37228 0.321536
\(679\) −4.74456 −0.182080
\(680\) 0 0
\(681\) −11.1168 −0.425998
\(682\) 6.37228 0.244007
\(683\) −15.8614 −0.606920 −0.303460 0.952844i \(-0.598142\pi\)
−0.303460 + 0.952844i \(0.598142\pi\)
\(684\) −6.37228 −0.243650
\(685\) 0 0
\(686\) 19.8614 0.758312
\(687\) −21.1168 −0.805658
\(688\) −6.37228 −0.242941
\(689\) 8.74456 0.333141
\(690\) 0 0
\(691\) −47.8614 −1.82073 −0.910367 0.413802i \(-0.864201\pi\)
−0.910367 + 0.413802i \(0.864201\pi\)
\(692\) 18.0000 0.684257
\(693\) 15.1168 0.574241
\(694\) −32.4674 −1.23244
\(695\) 0 0
\(696\) 2.74456 0.104032
\(697\) 72.4674 2.74490
\(698\) −18.7446 −0.709492
\(699\) 13.8614 0.524287
\(700\) 0 0
\(701\) 5.39403 0.203730 0.101865 0.994798i \(-0.467519\pi\)
0.101865 + 0.994798i \(0.467519\pi\)
\(702\) 2.00000 0.0754851
\(703\) 68.4674 2.58230
\(704\) 6.37228 0.240164
\(705\) 0 0
\(706\) −7.48913 −0.281857
\(707\) −21.6277 −0.813394
\(708\) 8.74456 0.328641
\(709\) 22.8832 0.859395 0.429697 0.902973i \(-0.358620\pi\)
0.429697 + 0.902973i \(0.358620\pi\)
\(710\) 0 0
\(711\) −10.3723 −0.388991
\(712\) −4.37228 −0.163858
\(713\) −2.37228 −0.0888426
\(714\) −16.0000 −0.598785
\(715\) 0 0
\(716\) −12.0000 −0.448461
\(717\) 6.51087 0.243153
\(718\) 2.37228 0.0885328
\(719\) 27.2554 1.01646 0.508228 0.861222i \(-0.330301\pi\)
0.508228 + 0.861222i \(0.330301\pi\)
\(720\) 0 0
\(721\) 18.9783 0.706787
\(722\) −21.6060 −0.804091
\(723\) −27.4891 −1.02233
\(724\) −13.8614 −0.515155
\(725\) 0 0
\(726\) 29.6060 1.09878
\(727\) −13.6277 −0.505424 −0.252712 0.967542i \(-0.581323\pi\)
−0.252712 + 0.967542i \(0.581323\pi\)
\(728\) −4.74456 −0.175845
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) 42.9783 1.58961
\(732\) 11.4891 0.424650
\(733\) 34.0000 1.25582 0.627909 0.778287i \(-0.283911\pi\)
0.627909 + 0.778287i \(0.283911\pi\)
\(734\) 14.2337 0.525375
\(735\) 0 0
\(736\) −2.37228 −0.0874434
\(737\) 4.74456 0.174768
\(738\) 10.7446 0.395513
\(739\) 22.9783 0.845269 0.422634 0.906300i \(-0.361105\pi\)
0.422634 + 0.906300i \(0.361105\pi\)
\(740\) 0 0
\(741\) 12.7446 0.468183
\(742\) −10.3723 −0.380778
\(743\) 29.6277 1.08694 0.543468 0.839430i \(-0.317111\pi\)
0.543468 + 0.839430i \(0.317111\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 0 0
\(746\) −13.8614 −0.507502
\(747\) 12.0000 0.439057
\(748\) −42.9783 −1.57144
\(749\) −15.1168 −0.552357
\(750\) 0 0
\(751\) −27.2554 −0.994565 −0.497283 0.867589i \(-0.665669\pi\)
−0.497283 + 0.867589i \(0.665669\pi\)
\(752\) −4.74456 −0.173016
\(753\) 4.00000 0.145768
\(754\) −5.48913 −0.199902
\(755\) 0 0
\(756\) −2.37228 −0.0862790
\(757\) 26.0000 0.944986 0.472493 0.881334i \(-0.343354\pi\)
0.472493 + 0.881334i \(0.343354\pi\)
\(758\) 0.138593 0.00503394
\(759\) −15.1168 −0.548707
\(760\) 0 0
\(761\) −18.1386 −0.657523 −0.328762 0.944413i \(-0.606631\pi\)
−0.328762 + 0.944413i \(0.606631\pi\)
\(762\) 9.48913 0.343755
\(763\) −16.0000 −0.579239
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −9.48913 −0.342856
\(767\) −17.4891 −0.631496
\(768\) −1.00000 −0.0360844
\(769\) −3.62772 −0.130819 −0.0654094 0.997859i \(-0.520835\pi\)
−0.0654094 + 0.997859i \(0.520835\pi\)
\(770\) 0 0
\(771\) 7.62772 0.274706
\(772\) 7.48913 0.269540
\(773\) 18.6060 0.669210 0.334605 0.942358i \(-0.391397\pi\)
0.334605 + 0.942358i \(0.391397\pi\)
\(774\) 6.37228 0.229047
\(775\) 0 0
\(776\) 2.00000 0.0717958
\(777\) 25.4891 0.914417
\(778\) 28.9783 1.03892
\(779\) 68.4674 2.45310
\(780\) 0 0
\(781\) −15.1168 −0.540923
\(782\) 16.0000 0.572159
\(783\) −2.74456 −0.0980827
\(784\) −1.37228 −0.0490100
\(785\) 0 0
\(786\) 18.2337 0.650374
\(787\) 27.1168 0.966611 0.483306 0.875452i \(-0.339436\pi\)
0.483306 + 0.875452i \(0.339436\pi\)
\(788\) 3.48913 0.124295
\(789\) 26.9783 0.960451
\(790\) 0 0
\(791\) 19.8614 0.706190
\(792\) −6.37228 −0.226429
\(793\) −22.9783 −0.815982
\(794\) 38.6060 1.37007
\(795\) 0 0
\(796\) −18.3723 −0.651188
\(797\) 43.4891 1.54046 0.770232 0.637764i \(-0.220140\pi\)
0.770232 + 0.637764i \(0.220140\pi\)
\(798\) −15.1168 −0.535130
\(799\) 32.0000 1.13208
\(800\) 0 0
\(801\) 4.37228 0.154487
\(802\) −28.3723 −1.00186
\(803\) −58.0951 −2.05013
\(804\) −0.744563 −0.0262587
\(805\) 0 0
\(806\) 2.00000 0.0704470
\(807\) 2.00000 0.0704033
\(808\) 9.11684 0.320729
\(809\) −30.6060 −1.07605 −0.538024 0.842929i \(-0.680829\pi\)
−0.538024 + 0.842929i \(0.680829\pi\)
\(810\) 0 0
\(811\) −46.3723 −1.62835 −0.814176 0.580619i \(-0.802811\pi\)
−0.814176 + 0.580619i \(0.802811\pi\)
\(812\) 6.51087 0.228487
\(813\) 31.1168 1.09132
\(814\) 68.4674 2.39978
\(815\) 0 0
\(816\) 6.74456 0.236107
\(817\) 40.6060 1.42062
\(818\) −24.2337 −0.847311
\(819\) 4.74456 0.165788
\(820\) 0 0
\(821\) 26.7446 0.933392 0.466696 0.884418i \(-0.345444\pi\)
0.466696 + 0.884418i \(0.345444\pi\)
\(822\) −19.4891 −0.679761
\(823\) 28.7446 1.00197 0.500986 0.865455i \(-0.332971\pi\)
0.500986 + 0.865455i \(0.332971\pi\)
\(824\) −8.00000 −0.278693
\(825\) 0 0
\(826\) 20.7446 0.721796
\(827\) −5.48913 −0.190876 −0.0954378 0.995435i \(-0.530425\pi\)
−0.0954378 + 0.995435i \(0.530425\pi\)
\(828\) 2.37228 0.0824425
\(829\) 32.0951 1.11471 0.557354 0.830275i \(-0.311816\pi\)
0.557354 + 0.830275i \(0.311816\pi\)
\(830\) 0 0
\(831\) 26.7446 0.927759
\(832\) 2.00000 0.0693375
\(833\) 9.25544 0.320682
\(834\) 0.744563 0.0257821
\(835\) 0 0
\(836\) −40.6060 −1.40439
\(837\) 1.00000 0.0345651
\(838\) 12.0000 0.414533
\(839\) −11.8614 −0.409501 −0.204751 0.978814i \(-0.565638\pi\)
−0.204751 + 0.978814i \(0.565638\pi\)
\(840\) 0 0
\(841\) −21.4674 −0.740254
\(842\) 3.48913 0.120243
\(843\) −5.25544 −0.181007
\(844\) −6.37228 −0.219343
\(845\) 0 0
\(846\) 4.74456 0.163121
\(847\) 70.2337 2.41326
\(848\) 4.37228 0.150145
\(849\) 12.0000 0.411839
\(850\) 0 0
\(851\) −25.4891 −0.873756
\(852\) 2.37228 0.0812730
\(853\) −24.0951 −0.825000 −0.412500 0.910958i \(-0.635344\pi\)
−0.412500 + 0.910958i \(0.635344\pi\)
\(854\) 27.2554 0.932662
\(855\) 0 0
\(856\) 6.37228 0.217800
\(857\) 15.4891 0.529098 0.264549 0.964372i \(-0.414777\pi\)
0.264549 + 0.964372i \(0.414777\pi\)
\(858\) 12.7446 0.435092
\(859\) 5.48913 0.187287 0.0936433 0.995606i \(-0.470149\pi\)
0.0936433 + 0.995606i \(0.470149\pi\)
\(860\) 0 0
\(861\) 25.4891 0.868667
\(862\) −18.9783 −0.646402
\(863\) −45.3505 −1.54375 −0.771875 0.635774i \(-0.780681\pi\)
−0.771875 + 0.635774i \(0.780681\pi\)
\(864\) 1.00000 0.0340207
\(865\) 0 0
\(866\) 18.8832 0.641676
\(867\) −28.4891 −0.967541
\(868\) −2.37228 −0.0805205
\(869\) −66.0951 −2.24212
\(870\) 0 0
\(871\) 1.48913 0.0504571
\(872\) 6.74456 0.228400
\(873\) −2.00000 −0.0676897
\(874\) 15.1168 0.511335
\(875\) 0 0
\(876\) 9.11684 0.308030
\(877\) −0.978251 −0.0330332 −0.0165166 0.999864i \(-0.505258\pi\)
−0.0165166 + 0.999864i \(0.505258\pi\)
\(878\) −3.25544 −0.109866
\(879\) 15.4891 0.522435
\(880\) 0 0
\(881\) −48.9783 −1.65012 −0.825060 0.565046i \(-0.808859\pi\)
−0.825060 + 0.565046i \(0.808859\pi\)
\(882\) 1.37228 0.0462071
\(883\) −16.1386 −0.543107 −0.271553 0.962423i \(-0.587537\pi\)
−0.271553 + 0.962423i \(0.587537\pi\)
\(884\) −13.4891 −0.453688
\(885\) 0 0
\(886\) −17.3505 −0.582903
\(887\) 19.2554 0.646534 0.323267 0.946308i \(-0.395219\pi\)
0.323267 + 0.946308i \(0.395219\pi\)
\(888\) −10.7446 −0.360564
\(889\) 22.5109 0.754991
\(890\) 0 0
\(891\) 6.37228 0.213479
\(892\) −20.7446 −0.694579
\(893\) 30.2337 1.01173
\(894\) 5.11684 0.171133
\(895\) 0 0
\(896\) −2.37228 −0.0792524
\(897\) −4.74456 −0.158416
\(898\) −36.9783 −1.23398
\(899\) −2.74456 −0.0915363
\(900\) 0 0
\(901\) −29.4891 −0.982425
\(902\) 68.4674 2.27971
\(903\) 15.1168 0.503057
\(904\) −8.37228 −0.278458
\(905\) 0 0
\(906\) −8.00000 −0.265782
\(907\) −5.48913 −0.182263 −0.0911317 0.995839i \(-0.529048\pi\)
−0.0911317 + 0.995839i \(0.529048\pi\)
\(908\) 11.1168 0.368925
\(909\) −9.11684 −0.302387
\(910\) 0 0
\(911\) −25.4891 −0.844492 −0.422246 0.906481i \(-0.638758\pi\)
−0.422246 + 0.906481i \(0.638758\pi\)
\(912\) 6.37228 0.211007
\(913\) 76.4674 2.53070
\(914\) −34.4674 −1.14008
\(915\) 0 0
\(916\) 21.1168 0.697720
\(917\) 43.2554 1.42842
\(918\) −6.74456 −0.222604
\(919\) 22.2337 0.733422 0.366711 0.930335i \(-0.380484\pi\)
0.366711 + 0.930335i \(0.380484\pi\)
\(920\) 0 0
\(921\) −14.9783 −0.493550
\(922\) −37.7228 −1.24233
\(923\) −4.74456 −0.156169
\(924\) −15.1168 −0.497308
\(925\) 0 0
\(926\) 25.4891 0.837625
\(927\) 8.00000 0.262754
\(928\) −2.74456 −0.0900947
\(929\) 1.11684 0.0366425 0.0183212 0.999832i \(-0.494168\pi\)
0.0183212 + 0.999832i \(0.494168\pi\)
\(930\) 0 0
\(931\) 8.74456 0.286591
\(932\) −13.8614 −0.454045
\(933\) −8.00000 −0.261908
\(934\) 29.4891 0.964914
\(935\) 0 0
\(936\) −2.00000 −0.0653720
\(937\) 42.7446 1.39640 0.698202 0.715901i \(-0.253984\pi\)
0.698202 + 0.715901i \(0.253984\pi\)
\(938\) −1.76631 −0.0576721
\(939\) 10.0000 0.326338
\(940\) 0 0
\(941\) 37.7228 1.22973 0.614864 0.788633i \(-0.289211\pi\)
0.614864 + 0.788633i \(0.289211\pi\)
\(942\) −3.62772 −0.118197
\(943\) −25.4891 −0.830040
\(944\) −8.74456 −0.284611
\(945\) 0 0
\(946\) 40.6060 1.32021
\(947\) −8.74456 −0.284160 −0.142080 0.989855i \(-0.545379\pi\)
−0.142080 + 0.989855i \(0.545379\pi\)
\(948\) 10.3723 0.336876
\(949\) −18.2337 −0.591891
\(950\) 0 0
\(951\) −13.2554 −0.429837
\(952\) 16.0000 0.518563
\(953\) 45.7228 1.48111 0.740554 0.671997i \(-0.234563\pi\)
0.740554 + 0.671997i \(0.234563\pi\)
\(954\) −4.37228 −0.141558
\(955\) 0 0
\(956\) −6.51087 −0.210577
\(957\) −17.4891 −0.565343
\(958\) 19.8614 0.641693
\(959\) −46.2337 −1.49296
\(960\) 0 0
\(961\) 1.00000 0.0322581
\(962\) 21.4891 0.692837
\(963\) −6.37228 −0.205344
\(964\) 27.4891 0.885365
\(965\) 0 0
\(966\) 5.62772 0.181069
\(967\) −8.00000 −0.257263 −0.128631 0.991692i \(-0.541058\pi\)
−0.128631 + 0.991692i \(0.541058\pi\)
\(968\) −29.6060 −0.951572
\(969\) −42.9783 −1.38066
\(970\) 0 0
\(971\) 54.7011 1.75544 0.877720 0.479173i \(-0.159063\pi\)
0.877720 + 0.479173i \(0.159063\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 1.76631 0.0566254
\(974\) −14.5109 −0.464958
\(975\) 0 0
\(976\) −11.4891 −0.367758
\(977\) 46.0000 1.47167 0.735835 0.677161i \(-0.236790\pi\)
0.735835 + 0.677161i \(0.236790\pi\)
\(978\) 10.2337 0.327237
\(979\) 27.8614 0.890454
\(980\) 0 0
\(981\) −6.74456 −0.215337
\(982\) 12.6060 0.402273
\(983\) −42.9783 −1.37079 −0.685397 0.728170i \(-0.740371\pi\)
−0.685397 + 0.728170i \(0.740371\pi\)
\(984\) −10.7446 −0.342524
\(985\) 0 0
\(986\) 18.5109 0.589506
\(987\) 11.2554 0.358265
\(988\) −12.7446 −0.405459
\(989\) −15.1168 −0.480688
\(990\) 0 0
\(991\) −7.39403 −0.234879 −0.117440 0.993080i \(-0.537469\pi\)
−0.117440 + 0.993080i \(0.537469\pi\)
\(992\) 1.00000 0.0317500
\(993\) −21.4891 −0.681937
\(994\) 5.62772 0.178500
\(995\) 0 0
\(996\) −12.0000 −0.380235
\(997\) −6.00000 −0.190022 −0.0950110 0.995476i \(-0.530289\pi\)
−0.0950110 + 0.995476i \(0.530289\pi\)
\(998\) −10.5109 −0.332716
\(999\) 10.7446 0.339943
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 4650.2.a.by.1.2 2
5.2 odd 4 4650.2.d.bh.3349.2 4
5.3 odd 4 4650.2.d.bh.3349.3 4
5.4 even 2 930.2.a.r.1.1 2
15.14 odd 2 2790.2.a.bd.1.1 2
20.19 odd 2 7440.2.a.bg.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.r.1.1 2 5.4 even 2
2790.2.a.bd.1.1 2 15.14 odd 2
4650.2.a.by.1.2 2 1.1 even 1 trivial
4650.2.d.bh.3349.2 4 5.2 odd 4
4650.2.d.bh.3349.3 4 5.3 odd 4
7440.2.a.bg.1.2 2 20.19 odd 2