Properties

Label 930.2.a.r.1.1
Level $930$
Weight $2$
Character 930.1
Self dual yes
Analytic conductor $7.426$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(7.42608738798\)
Analytic rank: \(0\)
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: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 930.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} +1.00000 q^{5} +1.00000 q^{6} -2.37228 q^{7} +1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{10} +6.37228 q^{11} +1.00000 q^{12} -2.00000 q^{13} -2.37228 q^{14} +1.00000 q^{15} +1.00000 q^{16} +6.74456 q^{17} +1.00000 q^{18} -6.37228 q^{19} +1.00000 q^{20} -2.37228 q^{21} +6.37228 q^{22} -2.37228 q^{23} +1.00000 q^{24} +1.00000 q^{25} -2.00000 q^{26} +1.00000 q^{27} -2.37228 q^{28} +2.74456 q^{29} +1.00000 q^{30} -1.00000 q^{31} +1.00000 q^{32} +6.37228 q^{33} +6.74456 q^{34} -2.37228 q^{35} +1.00000 q^{36} +10.7446 q^{37} -6.37228 q^{38} -2.00000 q^{39} +1.00000 q^{40} -10.7446 q^{41} -2.37228 q^{42} +6.37228 q^{43} +6.37228 q^{44} +1.00000 q^{45} -2.37228 q^{46} +4.74456 q^{47} +1.00000 q^{48} -1.37228 q^{49} +1.00000 q^{50} +6.74456 q^{51} -2.00000 q^{52} -4.37228 q^{53} +1.00000 q^{54} +6.37228 q^{55} -2.37228 q^{56} -6.37228 q^{57} +2.74456 q^{58} -8.74456 q^{59} +1.00000 q^{60} -11.4891 q^{61} -1.00000 q^{62} -2.37228 q^{63} +1.00000 q^{64} -2.00000 q^{65} +6.37228 q^{66} -0.744563 q^{67} +6.74456 q^{68} -2.37228 q^{69} -2.37228 q^{70} -2.37228 q^{71} +1.00000 q^{72} +9.11684 q^{73} +10.7446 q^{74} +1.00000 q^{75} -6.37228 q^{76} -15.1168 q^{77} -2.00000 q^{78} -10.3723 q^{79} +1.00000 q^{80} +1.00000 q^{81} -10.7446 q^{82} -12.0000 q^{83} -2.37228 q^{84} +6.74456 q^{85} +6.37228 q^{86} +2.74456 q^{87} +6.37228 q^{88} +4.37228 q^{89} +1.00000 q^{90} +4.74456 q^{91} -2.37228 q^{92} -1.00000 q^{93} +4.74456 q^{94} -6.37228 q^{95} +1.00000 q^{96} +2.00000 q^{97} -1.37228 q^{98} +6.37228 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{2} + 2 q^{3} + 2 q^{4} + 2 q^{5} + 2 q^{6} + q^{7} + 2 q^{8} + 2 q^{9} + 2 q^{10} + 7 q^{11} + 2 q^{12} - 4 q^{13} + q^{14} + 2 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 7 q^{19} + 2 q^{20} + q^{21} + 7 q^{22} + q^{23} + 2 q^{24} + 2 q^{25} - 4 q^{26} + 2 q^{27} + q^{28} - 6 q^{29} + 2 q^{30} - 2 q^{31} + 2 q^{32} + 7 q^{33} + 2 q^{34} + q^{35} + 2 q^{36} + 10 q^{37} - 7 q^{38} - 4 q^{39} + 2 q^{40} - 10 q^{41} + q^{42} + 7 q^{43} + 7 q^{44} + 2 q^{45} + q^{46} - 2 q^{47} + 2 q^{48} + 3 q^{49} + 2 q^{50} + 2 q^{51} - 4 q^{52} - 3 q^{53} + 2 q^{54} + 7 q^{55} + q^{56} - 7 q^{57} - 6 q^{58} - 6 q^{59} + 2 q^{60} - 2 q^{62} + q^{63} + 2 q^{64} - 4 q^{65} + 7 q^{66} + 10 q^{67} + 2 q^{68} + q^{69} + q^{70} + q^{71} + 2 q^{72} + q^{73} + 10 q^{74} + 2 q^{75} - 7 q^{76} - 13 q^{77} - 4 q^{78} - 15 q^{79} + 2 q^{80} + 2 q^{81} - 10 q^{82} - 24 q^{83} + q^{84} + 2 q^{85} + 7 q^{86} - 6 q^{87} + 7 q^{88} + 3 q^{89} + 2 q^{90} - 2 q^{91} + q^{92} - 2 q^{93} - 2 q^{94} - 7 q^{95} + 2 q^{96} + 4 q^{97} + 3 q^{98} + 7 q^{99}+O(q^{100}) \) 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
\(5\) 1.00000 0.447214
\(6\) 1.00000 0.408248
\(7\) −2.37228 −0.896638 −0.448319 0.893874i \(-0.647977\pi\)
−0.448319 + 0.893874i \(0.647977\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 1.00000 0.316228
\(11\) 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 \(-0.589456\pi\)
−0.277350 + 0.960769i \(0.589456\pi\)
\(14\) −2.37228 −0.634019
\(15\) 1.00000 0.258199
\(16\) 1.00000 0.250000
\(17\) 6.74456 1.63580 0.817898 0.575363i \(-0.195139\pi\)
0.817898 + 0.575363i \(0.195139\pi\)
\(18\) 1.00000 0.235702
\(19\) −6.37228 −1.46190 −0.730951 0.682430i \(-0.760923\pi\)
−0.730951 + 0.682430i \(0.760923\pi\)
\(20\) 1.00000 0.223607
\(21\) −2.37228 −0.517674
\(22\) 6.37228 1.35857
\(23\) −2.37228 −0.494655 −0.247327 0.968932i \(-0.579552\pi\)
−0.247327 + 0.968932i \(0.579552\pi\)
\(24\) 1.00000 0.204124
\(25\) 1.00000 0.200000
\(26\) −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 \(0.417982\pi\)
\(30\) 1.00000 0.182574
\(31\) −1.00000 −0.179605
\(32\) 1.00000 0.176777
\(33\) 6.37228 1.10927
\(34\) 6.74456 1.15668
\(35\) −2.37228 −0.400989
\(36\) 1.00000 0.166667
\(37\) 10.7446 1.76640 0.883198 0.469001i \(-0.155386\pi\)
0.883198 + 0.469001i \(0.155386\pi\)
\(38\) −6.37228 −1.03372
\(39\) −2.00000 −0.320256
\(40\) 1.00000 0.158114
\(41\) −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 \(-0.338499\pi\)
0.485882 + 0.874024i \(0.338499\pi\)
\(44\) 6.37228 0.960658
\(45\) 1.00000 0.149071
\(46\) −2.37228 −0.349774
\(47\) 4.74456 0.692066 0.346033 0.938222i \(-0.387529\pi\)
0.346033 + 0.938222i \(0.387529\pi\)
\(48\) 1.00000 0.144338
\(49\) −1.37228 −0.196040
\(50\) 1.00000 0.141421
\(51\) 6.74456 0.944428
\(52\) −2.00000 −0.277350
\(53\) −4.37228 −0.600579 −0.300290 0.953848i \(-0.597083\pi\)
−0.300290 + 0.953848i \(0.597083\pi\)
\(54\) 1.00000 0.136083
\(55\) 6.37228 0.859238
\(56\) −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 \(0.692756\pi\)
\(60\) 1.00000 0.129099
\(61\) −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
\(65\) −2.00000 −0.248069
\(66\) 6.37228 0.784374
\(67\) −0.744563 −0.0909628 −0.0454814 0.998965i \(-0.514482\pi\)
−0.0454814 + 0.998965i \(0.514482\pi\)
\(68\) 6.74456 0.817898
\(69\) −2.37228 −0.285589
\(70\) −2.37228 −0.283542
\(71\) −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 \(-0.320868\pi\)
0.533523 + 0.845786i \(0.320868\pi\)
\(74\) 10.7446 1.24903
\(75\) 1.00000 0.115470
\(76\) −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 \(0.698312\pi\)
\(80\) 1.00000 0.111803
\(81\) 1.00000 0.111111
\(82\) −10.7446 −1.18654
\(83\) −12.0000 −1.31717 −0.658586 0.752506i \(-0.728845\pi\)
−0.658586 + 0.752506i \(0.728845\pi\)
\(84\) −2.37228 −0.258837
\(85\) 6.74456 0.731551
\(86\) 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 \(0.425561\pi\)
\(90\) 1.00000 0.105409
\(91\) 4.74456 0.497365
\(92\) −2.37228 −0.247327
\(93\) −1.00000 −0.103695
\(94\) 4.74456 0.489364
\(95\) −6.37228 −0.653782
\(96\) 1.00000 0.102062
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) −1.37228 −0.138621
\(99\) 6.37228 0.640438
\(100\) 1.00000 0.100000
\(101\) −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 \(-0.628955\pi\)
−0.394132 + 0.919054i \(0.628955\pi\)
\(104\) −2.00000 −0.196116
\(105\) −2.37228 −0.231511
\(106\) −4.37228 −0.424674
\(107\) 6.37228 0.616032 0.308016 0.951381i \(-0.400335\pi\)
0.308016 + 0.951381i \(0.400335\pi\)
\(108\) 1.00000 0.0962250
\(109\) −6.74456 −0.646012 −0.323006 0.946397i \(-0.604693\pi\)
−0.323006 + 0.946397i \(0.604693\pi\)
\(110\) 6.37228 0.607573
\(111\) 10.7446 1.01983
\(112\) −2.37228 −0.224160
\(113\) −8.37228 −0.787598 −0.393799 0.919197i \(-0.628839\pi\)
−0.393799 + 0.919197i \(0.628839\pi\)
\(114\) −6.37228 −0.596819
\(115\) −2.37228 −0.221216
\(116\) 2.74456 0.254826
\(117\) −2.00000 −0.184900
\(118\) −8.74456 −0.805002
\(119\) −16.0000 −1.46672
\(120\) 1.00000 0.0912871
\(121\) 29.6060 2.69145
\(122\) −11.4891 −1.04018
\(123\) −10.7446 −0.968805
\(124\) −1.00000 −0.0898027
\(125\) 1.00000 0.0894427
\(126\) −2.37228 −0.211340
\(127\) −9.48913 −0.842024 −0.421012 0.907055i \(-0.638325\pi\)
−0.421012 + 0.907055i \(0.638325\pi\)
\(128\) 1.00000 0.0883883
\(129\) 6.37228 0.561048
\(130\) −2.00000 −0.175412
\(131\) 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
\(135\) 1.00000 0.0860663
\(136\) 6.74456 0.578341
\(137\) 19.4891 1.66507 0.832534 0.553974i \(-0.186889\pi\)
0.832534 + 0.553974i \(0.186889\pi\)
\(138\) −2.37228 −0.201942
\(139\) 0.744563 0.0631530 0.0315765 0.999501i \(-0.489947\pi\)
0.0315765 + 0.999501i \(0.489947\pi\)
\(140\) −2.37228 −0.200494
\(141\) 4.74456 0.399564
\(142\) −2.37228 −0.199077
\(143\) −12.7446 −1.06575
\(144\) 1.00000 0.0833333
\(145\) 2.74456 0.227924
\(146\) 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 \(0.432786\pi\)
\(150\) 1.00000 0.0816497
\(151\) −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
\(155\) −1.00000 −0.0803219
\(156\) −2.00000 −0.160128
\(157\) 3.62772 0.289523 0.144762 0.989467i \(-0.453758\pi\)
0.144762 + 0.989467i \(0.453758\pi\)
\(158\) −10.3723 −0.825174
\(159\) −4.37228 −0.346744
\(160\) 1.00000 0.0790569
\(161\) 5.62772 0.443526
\(162\) 1.00000 0.0785674
\(163\) −10.2337 −0.801564 −0.400782 0.916173i \(-0.631262\pi\)
−0.400782 + 0.916173i \(0.631262\pi\)
\(164\) −10.7446 −0.839009
\(165\) 6.37228 0.496081
\(166\) −12.0000 −0.931381
\(167\) −18.3723 −1.42169 −0.710845 0.703349i \(-0.751687\pi\)
−0.710845 + 0.703349i \(0.751687\pi\)
\(168\) −2.37228 −0.183025
\(169\) −9.00000 −0.692308
\(170\) 6.74456 0.517284
\(171\) −6.37228 −0.487301
\(172\) 6.37228 0.485882
\(173\) −18.0000 −1.36851 −0.684257 0.729241i \(-0.739873\pi\)
−0.684257 + 0.729241i \(0.739873\pi\)
\(174\) 2.74456 0.208065
\(175\) −2.37228 −0.179328
\(176\) 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 \(0.648028\pi\)
\(180\) 1.00000 0.0745356
\(181\) −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
\(185\) 10.7446 0.789956
\(186\) −1.00000 −0.0733236
\(187\) 42.9783 3.14288
\(188\) 4.74456 0.346033
\(189\) −2.37228 −0.172558
\(190\) −6.37228 −0.462294
\(191\) 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 \(-0.586871\pi\)
−0.269540 + 0.962989i \(0.586871\pi\)
\(194\) 2.00000 0.143592
\(195\) −2.00000 −0.143223
\(196\) −1.37228 −0.0980201
\(197\) −3.48913 −0.248590 −0.124295 0.992245i \(-0.539667\pi\)
−0.124295 + 0.992245i \(0.539667\pi\)
\(198\) 6.37228 0.452858
\(199\) −18.3723 −1.30238 −0.651188 0.758916i \(-0.725729\pi\)
−0.651188 + 0.758916i \(0.725729\pi\)
\(200\) 1.00000 0.0707107
\(201\) −0.744563 −0.0525174
\(202\) −9.11684 −0.641459
\(203\) −6.51087 −0.456974
\(204\) 6.74456 0.472214
\(205\) −10.7446 −0.750433
\(206\) −8.00000 −0.557386
\(207\) −2.37228 −0.164885
\(208\) −2.00000 −0.138675
\(209\) −40.6060 −2.80877
\(210\) −2.37228 −0.163703
\(211\) −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
\(215\) 6.37228 0.434586
\(216\) 1.00000 0.0680414
\(217\) 2.37228 0.161041
\(218\) −6.74456 −0.456799
\(219\) 9.11684 0.616059
\(220\) 6.37228 0.429619
\(221\) −13.4891 −0.907377
\(222\) 10.7446 0.721128
\(223\) 20.7446 1.38916 0.694579 0.719416i \(-0.255591\pi\)
0.694579 + 0.719416i \(0.255591\pi\)
\(224\) −2.37228 −0.158505
\(225\) 1.00000 0.0666667
\(226\) −8.37228 −0.556916
\(227\) −11.1168 −0.737851 −0.368925 0.929459i \(-0.620274\pi\)
−0.368925 + 0.929459i \(0.620274\pi\)
\(228\) −6.37228 −0.422015
\(229\) 21.1168 1.39544 0.697720 0.716370i \(-0.254198\pi\)
0.697720 + 0.716370i \(0.254198\pi\)
\(230\) −2.37228 −0.156424
\(231\) −15.1168 −0.994615
\(232\) 2.74456 0.180189
\(233\) 13.8614 0.908091 0.454045 0.890979i \(-0.349980\pi\)
0.454045 + 0.890979i \(0.349980\pi\)
\(234\) −2.00000 −0.130744
\(235\) 4.74456 0.309501
\(236\) −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 \(0.567534\pi\)
\(240\) 1.00000 0.0645497
\(241\) 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
\(245\) −1.37228 −0.0876718
\(246\) −10.7446 −0.685048
\(247\) 12.7446 0.810917
\(248\) −1.00000 −0.0635001
\(249\) −12.0000 −0.760469
\(250\) 1.00000 0.0632456
\(251\) −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
\(255\) 6.74456 0.422361
\(256\) 1.00000 0.0625000
\(257\) 7.62772 0.475804 0.237902 0.971289i \(-0.423540\pi\)
0.237902 + 0.971289i \(0.423540\pi\)
\(258\) 6.37228 0.396721
\(259\) −25.4891 −1.58382
\(260\) −2.00000 −0.124035
\(261\) 2.74456 0.169884
\(262\) 18.2337 1.12648
\(263\) 26.9783 1.66355 0.831775 0.555113i \(-0.187325\pi\)
0.831775 + 0.555113i \(0.187325\pi\)
\(264\) 6.37228 0.392187
\(265\) −4.37228 −0.268587
\(266\) 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 \(0.519420\pi\)
\(270\) 1.00000 0.0608581
\(271\) −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
\(275\) 6.37228 0.384263
\(276\) −2.37228 −0.142795
\(277\) 26.7446 1.60693 0.803463 0.595355i \(-0.202989\pi\)
0.803463 + 0.595355i \(0.202989\pi\)
\(278\) 0.744563 0.0446559
\(279\) −1.00000 −0.0598684
\(280\) −2.37228 −0.141771
\(281\) 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 \(-0.383914\pi\)
0.356663 + 0.934233i \(0.383914\pi\)
\(284\) −2.37228 −0.140769
\(285\) −6.37228 −0.377461
\(286\) −12.7446 −0.753602
\(287\) 25.4891 1.50458
\(288\) 1.00000 0.0589256
\(289\) 28.4891 1.67583
\(290\) 2.74456 0.161166
\(291\) 2.00000 0.117242
\(292\) 9.11684 0.533523
\(293\) 15.4891 0.904884 0.452442 0.891794i \(-0.350553\pi\)
0.452442 + 0.891794i \(0.350553\pi\)
\(294\) −1.37228 −0.0800331
\(295\) −8.74456 −0.509128
\(296\) 10.7446 0.624515
\(297\) 6.37228 0.369757
\(298\) 5.11684 0.296411
\(299\) 4.74456 0.274385
\(300\) 1.00000 0.0577350
\(301\) −15.1168 −0.871320
\(302\) −8.00000 −0.460348
\(303\) −9.11684 −0.523749
\(304\) −6.37228 −0.365475
\(305\) −11.4891 −0.657865
\(306\) 6.74456 0.385561
\(307\) −14.9783 −0.854854 −0.427427 0.904050i \(-0.640580\pi\)
−0.427427 + 0.904050i \(0.640580\pi\)
\(308\) −15.1168 −0.861362
\(309\) −8.00000 −0.455104
\(310\) −1.00000 −0.0567962
\(311\) 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 \(-0.408798\pi\)
0.282617 + 0.959233i \(0.408798\pi\)
\(314\) 3.62772 0.204724
\(315\) −2.37228 −0.133663
\(316\) −10.3723 −0.583486
\(317\) −13.2554 −0.744500 −0.372250 0.928133i \(-0.621414\pi\)
−0.372250 + 0.928133i \(0.621414\pi\)
\(318\) −4.37228 −0.245185
\(319\) 17.4891 0.979203
\(320\) 1.00000 0.0559017
\(321\) 6.37228 0.355666
\(322\) 5.62772 0.313621
\(323\) −42.9783 −2.39137
\(324\) 1.00000 0.0555556
\(325\) −2.00000 −0.110940
\(326\) −10.2337 −0.566792
\(327\) −6.74456 −0.372975
\(328\) −10.7446 −0.593269
\(329\) −11.2554 −0.620532
\(330\) 6.37228 0.350783
\(331\) 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
\(335\) −0.744563 −0.0406798
\(336\) −2.37228 −0.129419
\(337\) −16.9783 −0.924864 −0.462432 0.886655i \(-0.653023\pi\)
−0.462432 + 0.886655i \(0.653023\pi\)
\(338\) −9.00000 −0.489535
\(339\) −8.37228 −0.454720
\(340\) 6.74456 0.365775
\(341\) −6.37228 −0.345078
\(342\) −6.37228 −0.344574
\(343\) 19.8614 1.07242
\(344\) 6.37228 0.343570
\(345\) −2.37228 −0.127719
\(346\) −18.0000 −0.967686
\(347\) −32.4674 −1.74294 −0.871470 0.490449i \(-0.836833\pi\)
−0.871470 + 0.490449i \(0.836833\pi\)
\(348\) 2.74456 0.147124
\(349\) 18.7446 1.00337 0.501687 0.865049i \(-0.332713\pi\)
0.501687 + 0.865049i \(0.332713\pi\)
\(350\) −2.37228 −0.126804
\(351\) −2.00000 −0.106752
\(352\) 6.37228 0.339644
\(353\) −7.48913 −0.398606 −0.199303 0.979938i \(-0.563868\pi\)
−0.199303 + 0.979938i \(0.563868\pi\)
\(354\) −8.74456 −0.464768
\(355\) −2.37228 −0.125908
\(356\) 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 \(0.519940\pi\)
\(360\) 1.00000 0.0527046
\(361\) 21.6060 1.13716
\(362\) −13.8614 −0.728539
\(363\) 29.6060 1.55391
\(364\) 4.74456 0.248683
\(365\) 9.11684 0.477197
\(366\) −11.4891 −0.600546
\(367\) 14.2337 0.742992 0.371496 0.928434i \(-0.378845\pi\)
0.371496 + 0.928434i \(0.378845\pi\)
\(368\) −2.37228 −0.123664
\(369\) −10.7446 −0.559340
\(370\) 10.7446 0.558583
\(371\) 10.3723 0.538502
\(372\) −1.00000 −0.0518476
\(373\) −13.8614 −0.717716 −0.358858 0.933392i \(-0.616834\pi\)
−0.358858 + 0.933392i \(0.616834\pi\)
\(374\) 42.9783 2.22235
\(375\) 1.00000 0.0516398
\(376\) 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 \(0.501133\pi\)
\(380\) −6.37228 −0.326891
\(381\) −9.48913 −0.486143
\(382\) 16.0000 0.818631
\(383\) −9.48913 −0.484872 −0.242436 0.970167i \(-0.577946\pi\)
−0.242436 + 0.970167i \(0.577946\pi\)
\(384\) 1.00000 0.0510310
\(385\) −15.1168 −0.770426
\(386\) −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 \(0.762643\pi\)
\(390\) −2.00000 −0.101274
\(391\) −16.0000 −0.809155
\(392\) −1.37228 −0.0693107
\(393\) 18.2337 0.919768
\(394\) −3.48913 −0.175780
\(395\) −10.3723 −0.521886
\(396\) 6.37228 0.320219
\(397\) 38.6060 1.93758 0.968789 0.247887i \(-0.0797361\pi\)
0.968789 + 0.247887i \(0.0797361\pi\)
\(398\) −18.3723 −0.920919
\(399\) 15.1168 0.756789
\(400\) 1.00000 0.0500000
\(401\) 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
\(405\) 1.00000 0.0496904
\(406\) −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 \(0.295510\pi\)
\(410\) −10.7446 −0.530636
\(411\) 19.4891 0.961328
\(412\) −8.00000 −0.394132
\(413\) 20.7446 1.02077
\(414\) −2.37228 −0.116591
\(415\) −12.0000 −0.589057
\(416\) −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 \(0.594693\pi\)
\(420\) −2.37228 −0.115755
\(421\) −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
\(425\) 6.74456 0.327159
\(426\) −2.37228 −0.114937
\(427\) 27.2554 1.31898
\(428\) 6.37228 0.308016
\(429\) −12.7446 −0.615313
\(430\) 6.37228 0.307299
\(431\) 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 \(-0.350092\pi\)
0.453733 + 0.891138i \(0.350092\pi\)
\(434\) 2.37228 0.113873
\(435\) 2.74456 0.131592
\(436\) −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 \(0.475247\pi\)
\(440\) 6.37228 0.303787
\(441\) −1.37228 −0.0653467
\(442\) −13.4891 −0.641612
\(443\) −17.3505 −0.824349 −0.412174 0.911105i \(-0.635231\pi\)
−0.412174 + 0.911105i \(0.635231\pi\)
\(444\) 10.7446 0.509914
\(445\) 4.37228 0.207266
\(446\) 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 \(0.162461\pi\)
\(450\) 1.00000 0.0471405
\(451\) −68.4674 −3.22400
\(452\) −8.37228 −0.393799
\(453\) −8.00000 −0.375873
\(454\) −11.1168 −0.521739
\(455\) 4.74456 0.222429
\(456\) −6.37228 −0.298409
\(457\) −34.4674 −1.61232 −0.806158 0.591700i \(-0.798457\pi\)
−0.806158 + 0.591700i \(0.798457\pi\)
\(458\) 21.1168 0.986725
\(459\) 6.74456 0.314809
\(460\) −2.37228 −0.110608
\(461\) 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 \(-0.298224\pi\)
0.592290 + 0.805725i \(0.298224\pi\)
\(464\) 2.74456 0.127413
\(465\) −1.00000 −0.0463739
\(466\) 13.8614 0.642117
\(467\) 29.4891 1.36459 0.682297 0.731075i \(-0.260981\pi\)
0.682297 + 0.731075i \(0.260981\pi\)
\(468\) −2.00000 −0.0924500
\(469\) 1.76631 0.0815607
\(470\) 4.74456 0.218850
\(471\) 3.62772 0.167156
\(472\) −8.74456 −0.402501
\(473\) 40.6060 1.86706
\(474\) −10.3723 −0.476415
\(475\) −6.37228 −0.292380
\(476\) −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 \(0.649912\pi\)
\(480\) 1.00000 0.0456435
\(481\) −21.4891 −0.979820
\(482\) 27.4891 1.25210
\(483\) 5.62772 0.256070
\(484\) 29.6060 1.34573
\(485\) 2.00000 0.0908153
\(486\) 1.00000 0.0453609
\(487\) −14.5109 −0.657550 −0.328775 0.944408i \(-0.606636\pi\)
−0.328775 + 0.944408i \(0.606636\pi\)
\(488\) −11.4891 −0.520088
\(489\) −10.2337 −0.462783
\(490\) −1.37228 −0.0619934
\(491\) −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
\(495\) 6.37228 0.286413
\(496\) −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 \(0.424404\pi\)
\(500\) 1.00000 0.0447214
\(501\) −18.3723 −0.820813
\(502\) −4.00000 −0.178529
\(503\) −17.4891 −0.779802 −0.389901 0.920857i \(-0.627491\pi\)
−0.389901 + 0.920857i \(0.627491\pi\)
\(504\) −2.37228 −0.105670
\(505\) −9.11684 −0.405694
\(506\) −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 \(0.828703\pi\)
\(510\) 6.74456 0.298654
\(511\) −21.6277 −0.956754
\(512\) 1.00000 0.0441942
\(513\) −6.37228 −0.281343
\(514\) 7.62772 0.336444
\(515\) −8.00000 −0.352522
\(516\) 6.37228 0.280524
\(517\) 30.2337 1.32968
\(518\) −25.4891 −1.11993
\(519\) −18.0000 −0.790112
\(520\) −2.00000 −0.0877058
\(521\) −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 \(-0.387273\pi\)
0.346785 + 0.937944i \(0.387273\pi\)
\(524\) 18.2337 0.796542
\(525\) −2.37228 −0.103535
\(526\) 26.9783 1.17631
\(527\) −6.74456 −0.293798
\(528\) 6.37228 0.277318
\(529\) −17.3723 −0.755317
\(530\) −4.37228 −0.189920
\(531\) −8.74456 −0.379482
\(532\) 15.1168 0.655398
\(533\) 21.4891 0.930797
\(534\) 4.37228 0.189207
\(535\) 6.37228 0.275498
\(536\) −0.744563 −0.0321602
\(537\) −12.0000 −0.517838
\(538\) −2.00000 −0.0862261
\(539\) −8.74456 −0.376655
\(540\) 1.00000 0.0430331
\(541\) 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
\(545\) −6.74456 −0.288905
\(546\) 4.74456 0.203049
\(547\) 8.74456 0.373890 0.186945 0.982370i \(-0.440141\pi\)
0.186945 + 0.982370i \(0.440141\pi\)
\(548\) 19.4891 0.832534
\(549\) −11.4891 −0.490344
\(550\) 6.37228 0.271715
\(551\) −17.4891 −0.745062
\(552\) −2.37228 −0.100971
\(553\) 24.6060 1.04635
\(554\) 26.7446 1.13627
\(555\) 10.7446 0.456081
\(556\) 0.744563 0.0315765
\(557\) 29.1168 1.23372 0.616860 0.787073i \(-0.288404\pi\)
0.616860 + 0.787073i \(0.288404\pi\)
\(558\) −1.00000 −0.0423334
\(559\) −12.7446 −0.539038
\(560\) −2.37228 −0.100247
\(561\) 42.9783 1.81454
\(562\) 5.25544 0.221687
\(563\) 38.9783 1.64274 0.821369 0.570398i \(-0.193211\pi\)
0.821369 + 0.570398i \(0.193211\pi\)
\(564\) 4.74456 0.199782
\(565\) −8.37228 −0.352225
\(566\) 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 \(0.470787\pi\)
\(570\) −6.37228 −0.266905
\(571\) −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
\(575\) −2.37228 −0.0989310
\(576\) 1.00000 0.0416667
\(577\) 32.2337 1.34191 0.670953 0.741500i \(-0.265885\pi\)
0.670953 + 0.741500i \(0.265885\pi\)
\(578\) 28.4891 1.18499
\(579\) −7.48913 −0.311237
\(580\) 2.74456 0.113962
\(581\) 28.4674 1.18103
\(582\) 2.00000 0.0829027
\(583\) −27.8614 −1.15390
\(584\) 9.11684 0.377258
\(585\) −2.00000 −0.0826898
\(586\) 15.4891 0.639850
\(587\) 12.0000 0.495293 0.247647 0.968850i \(-0.420343\pi\)
0.247647 + 0.968850i \(0.420343\pi\)
\(588\) −1.37228 −0.0565919
\(589\) 6.37228 0.262565
\(590\) −8.74456 −0.360008
\(591\) −3.48913 −0.143523
\(592\) 10.7446 0.441599
\(593\) −0.978251 −0.0401719 −0.0200860 0.999798i \(-0.506394\pi\)
−0.0200860 + 0.999798i \(0.506394\pi\)
\(594\) 6.37228 0.261458
\(595\) −16.0000 −0.655936
\(596\) 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 \(0.546445\pi\)
\(600\) 1.00000 0.0408248
\(601\) 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
\(605\) 29.6060 1.20365
\(606\) −9.11684 −0.370346
\(607\) −10.3723 −0.420998 −0.210499 0.977594i \(-0.567509\pi\)
−0.210499 + 0.977594i \(0.567509\pi\)
\(608\) −6.37228 −0.258430
\(609\) −6.51087 −0.263834
\(610\) −11.4891 −0.465181
\(611\) −9.48913 −0.383889
\(612\) 6.74456 0.272633
\(613\) 12.5109 0.505309 0.252655 0.967557i \(-0.418696\pi\)
0.252655 + 0.967557i \(0.418696\pi\)
\(614\) −14.9783 −0.604473
\(615\) −10.7446 −0.433263
\(616\) −15.1168 −0.609075
\(617\) 25.1168 1.01117 0.505583 0.862778i \(-0.331277\pi\)
0.505583 + 0.862778i \(0.331277\pi\)
\(618\) −8.00000 −0.321807
\(619\) −18.2337 −0.732874 −0.366437 0.930443i \(-0.619422\pi\)
−0.366437 + 0.930443i \(0.619422\pi\)
\(620\) −1.00000 −0.0401610
\(621\) −2.37228 −0.0951964
\(622\) 8.00000 0.320771
\(623\) −10.3723 −0.415557
\(624\) −2.00000 −0.0800641
\(625\) 1.00000 0.0400000
\(626\) 10.0000 0.399680
\(627\) −40.6060 −1.62165
\(628\) 3.62772 0.144762
\(629\) 72.4674 2.88946
\(630\) −2.37228 −0.0945140
\(631\) 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
\(635\) −9.48913 −0.376564
\(636\) −4.37228 −0.173372
\(637\) 2.74456 0.108744
\(638\) 17.4891 0.692401
\(639\) −2.37228 −0.0938460
\(640\) 1.00000 0.0395285
\(641\) 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 \(-0.229494\pi\)
0.751162 + 0.660118i \(0.229494\pi\)
\(644\) 5.62772 0.221763
\(645\) 6.37228 0.250908
\(646\) −42.9783 −1.69096
\(647\) −35.5842 −1.39896 −0.699480 0.714652i \(-0.746585\pi\)
−0.699480 + 0.714652i \(0.746585\pi\)
\(648\) 1.00000 0.0392837
\(649\) −55.7228 −2.18731
\(650\) −2.00000 −0.0784465
\(651\) 2.37228 0.0929770
\(652\) −10.2337 −0.400782
\(653\) −4.97825 −0.194814 −0.0974070 0.995245i \(-0.531055\pi\)
−0.0974070 + 0.995245i \(0.531055\pi\)
\(654\) −6.74456 −0.263733
\(655\) 18.2337 0.712449
\(656\) −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 \(0.252657\pi\)
\(660\) 6.37228 0.248041
\(661\) 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
\(665\) 15.1168 0.586206
\(666\) 10.7446 0.416343
\(667\) −6.51087 −0.252102
\(668\) −18.3723 −0.710845
\(669\) 20.7446 0.802031
\(670\) −0.744563 −0.0287650
\(671\) −73.2119 −2.82632
\(672\) −2.37228 −0.0915127
\(673\) −23.4891 −0.905439 −0.452720 0.891653i \(-0.649546\pi\)
−0.452720 + 0.891653i \(0.649546\pi\)
\(674\) −16.9783 −0.653978
\(675\) 1.00000 0.0384900
\(676\) −9.00000 −0.346154
\(677\) −1.39403 −0.0535770 −0.0267885 0.999641i \(-0.508528\pi\)
−0.0267885 + 0.999641i \(0.508528\pi\)
\(678\) −8.37228 −0.321536
\(679\) −4.74456 −0.182080
\(680\) 6.74456 0.258642
\(681\) −11.1168 −0.425998
\(682\) −6.37228 −0.244007
\(683\) 15.8614 0.606920 0.303460 0.952844i \(-0.401858\pi\)
0.303460 + 0.952844i \(0.401858\pi\)
\(684\) −6.37228 −0.243650
\(685\) 19.4891 0.744641
\(686\) 19.8614 0.758312
\(687\) 21.1168 0.805658
\(688\) 6.37228 0.242941
\(689\) 8.74456 0.333141
\(690\) −2.37228 −0.0903112
\(691\) −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
\(695\) 0.744563 0.0282429
\(696\) 2.74456 0.104032
\(697\) −72.4674 −2.74490
\(698\) 18.7446 0.709492
\(699\) 13.8614 0.524287
\(700\) −2.37228 −0.0896638
\(701\) 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
\(705\) 4.74456 0.178691
\(706\) −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 \(0.358620\pi\)
\(710\) −2.37228 −0.0890301
\(711\) −10.3723 −0.388991
\(712\) 4.37228 0.163858
\(713\) 2.37228 0.0888426
\(714\) −16.0000 −0.598785
\(715\) −12.7446 −0.476620
\(716\) −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 \(0.330301\pi\)
\(720\) 1.00000 0.0372678
\(721\) 18.9783 0.706787
\(722\) 21.6060 0.804091
\(723\) 27.4891 1.02233
\(724\) −13.8614 −0.515155
\(725\) 2.74456 0.101930
\(726\) 29.6060 1.09878
\(727\) 13.6277 0.505424 0.252712 0.967542i \(-0.418677\pi\)
0.252712 + 0.967542i \(0.418677\pi\)
\(728\) 4.74456 0.175845
\(729\) 1.00000 0.0370370
\(730\) 9.11684 0.337430
\(731\) 42.9783 1.58961
\(732\) −11.4891 −0.424650
\(733\) −34.0000 −1.25582 −0.627909 0.778287i \(-0.716089\pi\)
−0.627909 + 0.778287i \(0.716089\pi\)
\(734\) 14.2337 0.525375
\(735\) −1.37228 −0.0506174
\(736\) −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 \(0.361105\pi\)
\(740\) 10.7446 0.394978
\(741\) 12.7446 0.468183
\(742\) 10.3723 0.380778
\(743\) −29.6277 −1.08694 −0.543468 0.839430i \(-0.682889\pi\)
−0.543468 + 0.839430i \(0.682889\pi\)
\(744\) −1.00000 −0.0366618
\(745\) 5.11684 0.187467
\(746\) −13.8614 −0.507502
\(747\) −12.0000 −0.439057
\(748\) 42.9783 1.57144
\(749\) −15.1168 −0.552357
\(750\) 1.00000 0.0365148
\(751\) −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
\(755\) −8.00000 −0.291150
\(756\) −2.37228 −0.0862790
\(757\) −26.0000 −0.944986 −0.472493 0.881334i \(-0.656646\pi\)
−0.472493 + 0.881334i \(0.656646\pi\)
\(758\) −0.138593 −0.00503394
\(759\) −15.1168 −0.548707
\(760\) −6.37228 −0.231147
\(761\) −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
\(765\) 6.74456 0.243850
\(766\) −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 \(0.520835\pi\)
\(770\) −15.1168 −0.544773
\(771\) 7.62772 0.274706
\(772\) −7.48913 −0.269540
\(773\) −18.6060 −0.669210 −0.334605 0.942358i \(-0.608603\pi\)
−0.334605 + 0.942358i \(0.608603\pi\)
\(774\) 6.37228 0.229047
\(775\) −1.00000 −0.0359211
\(776\) 2.00000 0.0717958
\(777\) −25.4891 −0.914417
\(778\) −28.9783 −1.03892
\(779\) 68.4674 2.45310
\(780\) −2.00000 −0.0716115
\(781\) −15.1168 −0.540923
\(782\) −16.0000 −0.572159
\(783\) 2.74456 0.0980827
\(784\) −1.37228 −0.0490100
\(785\) 3.62772 0.129479
\(786\) 18.2337 0.650374
\(787\) −27.1168 −0.966611 −0.483306 0.875452i \(-0.660564\pi\)
−0.483306 + 0.875452i \(0.660564\pi\)
\(788\) −3.48913 −0.124295
\(789\) 26.9783 0.960451
\(790\) −10.3723 −0.369029
\(791\) 19.8614 0.706190
\(792\) 6.37228 0.226429
\(793\) 22.9783 0.815982
\(794\) 38.6060 1.37007
\(795\) −4.37228 −0.155069
\(796\) −18.3723 −0.651188
\(797\) −43.4891 −1.54046 −0.770232 0.637764i \(-0.779860\pi\)
−0.770232 + 0.637764i \(0.779860\pi\)
\(798\) 15.1168 0.535130
\(799\) 32.0000 1.13208
\(800\) 1.00000 0.0353553
\(801\) 4.37228 0.154487
\(802\) 28.3723 1.00186
\(803\) 58.0951 2.05013
\(804\) −0.744563 −0.0262587
\(805\) 5.62772 0.198351
\(806\) 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 \(0.680829\pi\)
\(810\) 1.00000 0.0351364
\(811\) −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
\(815\) −10.2337 −0.358470
\(816\) 6.74456 0.236107
\(817\) −40.6060 −1.42062
\(818\) 24.2337 0.847311
\(819\) 4.74456 0.165788
\(820\) −10.7446 −0.375216
\(821\) 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 \(-0.667029\pi\)
−0.500986 + 0.865455i \(0.667029\pi\)
\(824\) −8.00000 −0.278693
\(825\) 6.37228 0.221854
\(826\) 20.7446 0.721796
\(827\) 5.48913 0.190876 0.0954378 0.995435i \(-0.469575\pi\)
0.0954378 + 0.995435i \(0.469575\pi\)
\(828\) −2.37228 −0.0824425
\(829\) 32.0951 1.11471 0.557354 0.830275i \(-0.311816\pi\)
0.557354 + 0.830275i \(0.311816\pi\)
\(830\) −12.0000 −0.416526
\(831\) 26.7446 0.927759
\(832\) −2.00000 −0.0693375
\(833\) −9.25544 −0.320682
\(834\) 0.744563 0.0257821
\(835\) −18.3723 −0.635799
\(836\) −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 \(0.565638\pi\)
\(840\) −2.37228 −0.0818515
\(841\) −21.4674 −0.740254
\(842\) −3.48913 −0.120243
\(843\) 5.25544 0.181007
\(844\) −6.37228 −0.219343
\(845\) −9.00000 −0.309609
\(846\) 4.74456 0.163121
\(847\) −70.2337 −2.41326
\(848\) −4.37228 −0.150145
\(849\) 12.0000 0.411839
\(850\) 6.74456 0.231337
\(851\) −25.4891 −0.873756
\(852\) −2.37228 −0.0812730
\(853\) 24.0951 0.825000 0.412500 0.910958i \(-0.364656\pi\)
0.412500 + 0.910958i \(0.364656\pi\)
\(854\) 27.2554 0.932662
\(855\) −6.37228 −0.217927
\(856\) 6.37228 0.217800
\(857\) −15.4891 −0.529098 −0.264549 0.964372i \(-0.585223\pi\)
−0.264549 + 0.964372i \(0.585223\pi\)
\(858\) −12.7446 −0.435092
\(859\) 5.48913 0.187287 0.0936433 0.995606i \(-0.470149\pi\)
0.0936433 + 0.995606i \(0.470149\pi\)
\(860\) 6.37228 0.217293
\(861\) 25.4891 0.868667
\(862\) 18.9783 0.646402
\(863\) 45.3505 1.54375 0.771875 0.635774i \(-0.219319\pi\)
0.771875 + 0.635774i \(0.219319\pi\)
\(864\) 1.00000 0.0340207
\(865\) −18.0000 −0.612018
\(866\) 18.8832 0.641676
\(867\) 28.4891 0.967541
\(868\) 2.37228 0.0805205
\(869\) −66.0951 −2.24212
\(870\) 2.74456 0.0930494
\(871\) 1.48913 0.0504571
\(872\) −6.74456 −0.228400
\(873\) 2.00000 0.0676897
\(874\) 15.1168 0.511335
\(875\) −2.37228 −0.0801977
\(876\) 9.11684 0.308030
\(877\) 0.978251 0.0330332 0.0165166 0.999864i \(-0.494742\pi\)
0.0165166 + 0.999864i \(0.494742\pi\)
\(878\) 3.25544 0.109866
\(879\) 15.4891 0.522435
\(880\) 6.37228 0.214810
\(881\) −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 \(-0.412463\pi\)
0.271553 + 0.962423i \(0.412463\pi\)
\(884\) −13.4891 −0.453688
\(885\) −8.74456 −0.293945
\(886\) −17.3505 −0.582903
\(887\) −19.2554 −0.646534 −0.323267 0.946308i \(-0.604781\pi\)
−0.323267 + 0.946308i \(0.604781\pi\)
\(888\) 10.7446 0.360564
\(889\) 22.5109 0.754991
\(890\) 4.37228 0.146559
\(891\) 6.37228 0.213479
\(892\) 20.7446 0.694579
\(893\) −30.2337 −1.01173
\(894\) 5.11684 0.171133
\(895\) −12.0000 −0.401116
\(896\) −2.37228 −0.0792524
\(897\) 4.74456 0.158416
\(898\) 36.9783 1.23398
\(899\) −2.74456 −0.0915363
\(900\) 1.00000 0.0333333
\(901\) −29.4891 −0.982425
\(902\) −68.4674 −2.27971
\(903\) −15.1168 −0.503057
\(904\) −8.37228 −0.278458
\(905\) −13.8614 −0.460769
\(906\) −8.00000 −0.265782
\(907\) 5.48913 0.182263 0.0911317 0.995839i \(-0.470952\pi\)
0.0911317 + 0.995839i \(0.470952\pi\)
\(908\) −11.1168 −0.368925
\(909\) −9.11684 −0.302387
\(910\) 4.74456 0.157281
\(911\) −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
\(915\) −11.4891 −0.379819
\(916\) 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 \(0.380484\pi\)
\(920\) −2.37228 −0.0782118
\(921\) −14.9783 −0.493550
\(922\) 37.7228 1.24233
\(923\) 4.74456 0.156169
\(924\) −15.1168 −0.497308
\(925\) 10.7446 0.353279
\(926\) 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 \(0.494168\pi\)
\(930\) −1.00000 −0.0327913
\(931\) 8.74456 0.286591
\(932\) 13.8614 0.454045
\(933\) 8.00000 0.261908
\(934\) 29.4891 0.964914
\(935\) 42.9783 1.40554
\(936\) −2.00000 −0.0653720
\(937\) −42.7446 −1.39640 −0.698202 0.715901i \(-0.746016\pi\)
−0.698202 + 0.715901i \(0.746016\pi\)
\(938\) 1.76631 0.0576721
\(939\) 10.0000 0.326338
\(940\) 4.74456 0.154751
\(941\) 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
\(945\) −2.37228 −0.0771703
\(946\) 40.6060 1.32021
\(947\) 8.74456 0.284160 0.142080 0.989855i \(-0.454621\pi\)
0.142080 + 0.989855i \(0.454621\pi\)
\(948\) −10.3723 −0.336876
\(949\) −18.2337 −0.591891
\(950\) −6.37228 −0.206744
\(951\) −13.2554 −0.429837
\(952\) −16.0000 −0.518563
\(953\) −45.7228 −1.48111 −0.740554 0.671997i \(-0.765437\pi\)
−0.740554 + 0.671997i \(0.765437\pi\)
\(954\) −4.37228 −0.141558
\(955\) 16.0000 0.517748
\(956\) −6.51087 −0.210577
\(957\) 17.4891 0.565343
\(958\) −19.8614 −0.641693
\(959\) −46.2337 −1.49296
\(960\) 1.00000 0.0322749
\(961\) 1.00000 0.0322581
\(962\) −21.4891 −0.692837
\(963\) 6.37228 0.205344
\(964\) 27.4891 0.885365
\(965\) −7.48913 −0.241083
\(966\) 5.62772 0.181069
\(967\) 8.00000 0.257263 0.128631 0.991692i \(-0.458942\pi\)
0.128631 + 0.991692i \(0.458942\pi\)
\(968\) 29.6060 0.951572
\(969\) −42.9783 −1.38066
\(970\) 2.00000 0.0642161
\(971\) 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
\(975\) −2.00000 −0.0640513
\(976\) −11.4891 −0.367758
\(977\) −46.0000 −1.47167 −0.735835 0.677161i \(-0.763210\pi\)
−0.735835 + 0.677161i \(0.763210\pi\)
\(978\) −10.2337 −0.327237
\(979\) 27.8614 0.890454
\(980\) −1.37228 −0.0438359
\(981\) −6.74456 −0.215337
\(982\) −12.6060 −0.402273
\(983\) 42.9783 1.37079 0.685397 0.728170i \(-0.259629\pi\)
0.685397 + 0.728170i \(0.259629\pi\)
\(984\) −10.7446 −0.342524
\(985\) −3.48913 −0.111173
\(986\) 18.5109 0.589506
\(987\) −11.2554 −0.358265
\(988\) 12.7446 0.405459
\(989\) −15.1168 −0.480688
\(990\) 6.37228 0.202524
\(991\) −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
\(995\) −18.3723 −0.582440
\(996\) −12.0000 −0.380235
\(997\) 6.00000 0.190022 0.0950110 0.995476i \(-0.469711\pi\)
0.0950110 + 0.995476i \(0.469711\pi\)
\(998\) 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 930.2.a.r.1.1 2
3.2 odd 2 2790.2.a.bd.1.1 2
4.3 odd 2 7440.2.a.bg.1.2 2
5.2 odd 4 4650.2.d.bh.3349.3 4
5.3 odd 4 4650.2.d.bh.3349.2 4
5.4 even 2 4650.2.a.by.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
930.2.a.r.1.1 2 1.1 even 1 trivial
2790.2.a.bd.1.1 2 3.2 odd 2
4650.2.a.by.1.2 2 5.4 even 2
4650.2.d.bh.3349.2 4 5.3 odd 4
4650.2.d.bh.3349.3 4 5.2 odd 4
7440.2.a.bg.1.2 2 4.3 odd 2