Properties

Label 6930.2.a.bo.1.1
Level $6930$
Weight $2$
Character 6930.1
Self dual yes
Analytic conductor $55.336$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 6930 = 2 \cdot 3^{2} \cdot 5 \cdot 7 \cdot 11 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 6930.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(55.3363286007\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\sqrt{33}) \)
Defining polynomial: \(x^{2} - x - 8\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 770)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-2.37228\) of defining polynomial
Character \(\chi\) \(=\) 6930.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{7} -1.00000 q^{8} +1.00000 q^{10} +1.00000 q^{11} -4.74456 q^{13} -1.00000 q^{14} +1.00000 q^{16} -4.74456 q^{17} +4.74456 q^{19} -1.00000 q^{20} -1.00000 q^{22} -4.74456 q^{23} +1.00000 q^{25} +4.74456 q^{26} +1.00000 q^{28} +2.74456 q^{29} -6.74456 q^{31} -1.00000 q^{32} +4.74456 q^{34} -1.00000 q^{35} -10.7446 q^{37} -4.74456 q^{38} +1.00000 q^{40} +4.00000 q^{41} +4.00000 q^{43} +1.00000 q^{44} +4.74456 q^{46} +6.74456 q^{47} +1.00000 q^{49} -1.00000 q^{50} -4.74456 q^{52} +1.25544 q^{53} -1.00000 q^{55} -1.00000 q^{56} -2.74456 q^{58} -2.74456 q^{59} +12.7446 q^{61} +6.74456 q^{62} +1.00000 q^{64} +4.74456 q^{65} -4.00000 q^{67} -4.74456 q^{68} +1.00000 q^{70} +4.00000 q^{71} -0.744563 q^{73} +10.7446 q^{74} +4.74456 q^{76} +1.00000 q^{77} -4.74456 q^{79} -1.00000 q^{80} -4.00000 q^{82} -8.00000 q^{83} +4.74456 q^{85} -4.00000 q^{86} -1.00000 q^{88} +7.48913 q^{89} -4.74456 q^{91} -4.74456 q^{92} -6.74456 q^{94} -4.74456 q^{95} -5.25544 q^{97} -1.00000 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + O(q^{10}) \) \( 2q - 2q^{2} + 2q^{4} - 2q^{5} + 2q^{7} - 2q^{8} + 2q^{10} + 2q^{11} + 2q^{13} - 2q^{14} + 2q^{16} + 2q^{17} - 2q^{19} - 2q^{20} - 2q^{22} + 2q^{23} + 2q^{25} - 2q^{26} + 2q^{28} - 6q^{29} - 2q^{31} - 2q^{32} - 2q^{34} - 2q^{35} - 10q^{37} + 2q^{38} + 2q^{40} + 8q^{41} + 8q^{43} + 2q^{44} - 2q^{46} + 2q^{47} + 2q^{49} - 2q^{50} + 2q^{52} + 14q^{53} - 2q^{55} - 2q^{56} + 6q^{58} + 6q^{59} + 14q^{61} + 2q^{62} + 2q^{64} - 2q^{65} - 8q^{67} + 2q^{68} + 2q^{70} + 8q^{71} + 10q^{73} + 10q^{74} - 2q^{76} + 2q^{77} + 2q^{79} - 2q^{80} - 8q^{82} - 16q^{83} - 2q^{85} - 8q^{86} - 2q^{88} - 8q^{89} + 2q^{91} + 2q^{92} - 2q^{94} + 2q^{95} - 22q^{97} - 2q^{98} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.00000 −0.707107
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) −1.00000 −0.447214
\(6\) 0 0
\(7\) 1.00000 0.377964
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) 1.00000 0.301511
\(12\) 0 0
\(13\) −4.74456 −1.31590 −0.657952 0.753059i \(-0.728577\pi\)
−0.657952 + 0.753059i \(0.728577\pi\)
\(14\) −1.00000 −0.267261
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −4.74456 −1.15073 −0.575363 0.817898i \(-0.695139\pi\)
−0.575363 + 0.817898i \(0.695139\pi\)
\(18\) 0 0
\(19\) 4.74456 1.08848 0.544239 0.838930i \(-0.316819\pi\)
0.544239 + 0.838930i \(0.316819\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) −1.00000 −0.213201
\(23\) −4.74456 −0.989310 −0.494655 0.869090i \(-0.664706\pi\)
−0.494655 + 0.869090i \(0.664706\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 4.74456 0.930485
\(27\) 0 0
\(28\) 1.00000 0.188982
\(29\) 2.74456 0.509652 0.254826 0.966987i \(-0.417982\pi\)
0.254826 + 0.966987i \(0.417982\pi\)
\(30\) 0 0
\(31\) −6.74456 −1.21136 −0.605680 0.795709i \(-0.707099\pi\)
−0.605680 + 0.795709i \(0.707099\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 4.74456 0.813686
\(35\) −1.00000 −0.169031
\(36\) 0 0
\(37\) −10.7446 −1.76640 −0.883198 0.469001i \(-0.844614\pi\)
−0.883198 + 0.469001i \(0.844614\pi\)
\(38\) −4.74456 −0.769670
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 4.00000 0.624695 0.312348 0.949968i \(-0.398885\pi\)
0.312348 + 0.949968i \(0.398885\pi\)
\(42\) 0 0
\(43\) 4.00000 0.609994 0.304997 0.952353i \(-0.401344\pi\)
0.304997 + 0.952353i \(0.401344\pi\)
\(44\) 1.00000 0.150756
\(45\) 0 0
\(46\) 4.74456 0.699548
\(47\) 6.74456 0.983796 0.491898 0.870653i \(-0.336303\pi\)
0.491898 + 0.870653i \(0.336303\pi\)
\(48\) 0 0
\(49\) 1.00000 0.142857
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −4.74456 −0.657952
\(53\) 1.25544 0.172448 0.0862238 0.996276i \(-0.472520\pi\)
0.0862238 + 0.996276i \(0.472520\pi\)
\(54\) 0 0
\(55\) −1.00000 −0.134840
\(56\) −1.00000 −0.133631
\(57\) 0 0
\(58\) −2.74456 −0.360379
\(59\) −2.74456 −0.357312 −0.178656 0.983912i \(-0.557175\pi\)
−0.178656 + 0.983912i \(0.557175\pi\)
\(60\) 0 0
\(61\) 12.7446 1.63177 0.815887 0.578211i \(-0.196249\pi\)
0.815887 + 0.578211i \(0.196249\pi\)
\(62\) 6.74456 0.856560
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 4.74456 0.588491
\(66\) 0 0
\(67\) −4.00000 −0.488678 −0.244339 0.969690i \(-0.578571\pi\)
−0.244339 + 0.969690i \(0.578571\pi\)
\(68\) −4.74456 −0.575363
\(69\) 0 0
\(70\) 1.00000 0.119523
\(71\) 4.00000 0.474713 0.237356 0.971423i \(-0.423719\pi\)
0.237356 + 0.971423i \(0.423719\pi\)
\(72\) 0 0
\(73\) −0.744563 −0.0871445 −0.0435722 0.999050i \(-0.513874\pi\)
−0.0435722 + 0.999050i \(0.513874\pi\)
\(74\) 10.7446 1.24903
\(75\) 0 0
\(76\) 4.74456 0.544239
\(77\) 1.00000 0.113961
\(78\) 0 0
\(79\) −4.74456 −0.533805 −0.266903 0.963724i \(-0.586000\pi\)
−0.266903 + 0.963724i \(0.586000\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −4.00000 −0.441726
\(83\) −8.00000 −0.878114 −0.439057 0.898459i \(-0.644687\pi\)
−0.439057 + 0.898459i \(0.644687\pi\)
\(84\) 0 0
\(85\) 4.74456 0.514620
\(86\) −4.00000 −0.431331
\(87\) 0 0
\(88\) −1.00000 −0.106600
\(89\) 7.48913 0.793846 0.396923 0.917852i \(-0.370078\pi\)
0.396923 + 0.917852i \(0.370078\pi\)
\(90\) 0 0
\(91\) −4.74456 −0.497365
\(92\) −4.74456 −0.494655
\(93\) 0 0
\(94\) −6.74456 −0.695649
\(95\) −4.74456 −0.486782
\(96\) 0 0
\(97\) −5.25544 −0.533609 −0.266804 0.963751i \(-0.585968\pi\)
−0.266804 + 0.963751i \(0.585968\pi\)
\(98\) −1.00000 −0.101015
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 8.74456 0.870117 0.435058 0.900402i \(-0.356728\pi\)
0.435058 + 0.900402i \(0.356728\pi\)
\(102\) 0 0
\(103\) 10.7446 1.05869 0.529347 0.848406i \(-0.322437\pi\)
0.529347 + 0.848406i \(0.322437\pi\)
\(104\) 4.74456 0.465243
\(105\) 0 0
\(106\) −1.25544 −0.121939
\(107\) 12.0000 1.16008 0.580042 0.814587i \(-0.303036\pi\)
0.580042 + 0.814587i \(0.303036\pi\)
\(108\) 0 0
\(109\) 16.2337 1.55491 0.777453 0.628941i \(-0.216511\pi\)
0.777453 + 0.628941i \(0.216511\pi\)
\(110\) 1.00000 0.0953463
\(111\) 0 0
\(112\) 1.00000 0.0944911
\(113\) −10.0000 −0.940721 −0.470360 0.882474i \(-0.655876\pi\)
−0.470360 + 0.882474i \(0.655876\pi\)
\(114\) 0 0
\(115\) 4.74456 0.442433
\(116\) 2.74456 0.254826
\(117\) 0 0
\(118\) 2.74456 0.252657
\(119\) −4.74456 −0.434933
\(120\) 0 0
\(121\) 1.00000 0.0909091
\(122\) −12.7446 −1.15384
\(123\) 0 0
\(124\) −6.74456 −0.605680
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −4.74456 −0.416126
\(131\) −8.74456 −0.764016 −0.382008 0.924159i \(-0.624767\pi\)
−0.382008 + 0.924159i \(0.624767\pi\)
\(132\) 0 0
\(133\) 4.74456 0.411406
\(134\) 4.00000 0.345547
\(135\) 0 0
\(136\) 4.74456 0.406843
\(137\) 19.4891 1.66507 0.832534 0.553974i \(-0.186889\pi\)
0.832534 + 0.553974i \(0.186889\pi\)
\(138\) 0 0
\(139\) 3.25544 0.276123 0.138061 0.990424i \(-0.455913\pi\)
0.138061 + 0.990424i \(0.455913\pi\)
\(140\) −1.00000 −0.0845154
\(141\) 0 0
\(142\) −4.00000 −0.335673
\(143\) −4.74456 −0.396760
\(144\) 0 0
\(145\) −2.74456 −0.227924
\(146\) 0.744563 0.0616204
\(147\) 0 0
\(148\) −10.7446 −0.883198
\(149\) −10.7446 −0.880229 −0.440114 0.897942i \(-0.645062\pi\)
−0.440114 + 0.897942i \(0.645062\pi\)
\(150\) 0 0
\(151\) −20.7446 −1.68817 −0.844084 0.536211i \(-0.819855\pi\)
−0.844084 + 0.536211i \(0.819855\pi\)
\(152\) −4.74456 −0.384835
\(153\) 0 0
\(154\) −1.00000 −0.0805823
\(155\) 6.74456 0.541736
\(156\) 0 0
\(157\) 23.4891 1.87464 0.937318 0.348475i \(-0.113300\pi\)
0.937318 + 0.348475i \(0.113300\pi\)
\(158\) 4.74456 0.377457
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −4.74456 −0.373924
\(162\) 0 0
\(163\) −4.00000 −0.313304 −0.156652 0.987654i \(-0.550070\pi\)
−0.156652 + 0.987654i \(0.550070\pi\)
\(164\) 4.00000 0.312348
\(165\) 0 0
\(166\) 8.00000 0.620920
\(167\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(168\) 0 0
\(169\) 9.51087 0.731606
\(170\) −4.74456 −0.363891
\(171\) 0 0
\(172\) 4.00000 0.304997
\(173\) 14.2337 1.08217 0.541084 0.840969i \(-0.318014\pi\)
0.541084 + 0.840969i \(0.318014\pi\)
\(174\) 0 0
\(175\) 1.00000 0.0755929
\(176\) 1.00000 0.0753778
\(177\) 0 0
\(178\) −7.48913 −0.561334
\(179\) 4.00000 0.298974 0.149487 0.988764i \(-0.452238\pi\)
0.149487 + 0.988764i \(0.452238\pi\)
\(180\) 0 0
\(181\) 3.48913 0.259345 0.129672 0.991557i \(-0.458607\pi\)
0.129672 + 0.991557i \(0.458607\pi\)
\(182\) 4.74456 0.351690
\(183\) 0 0
\(184\) 4.74456 0.349774
\(185\) 10.7446 0.789956
\(186\) 0 0
\(187\) −4.74456 −0.346957
\(188\) 6.74456 0.491898
\(189\) 0 0
\(190\) 4.74456 0.344207
\(191\) 16.0000 1.15772 0.578860 0.815427i \(-0.303498\pi\)
0.578860 + 0.815427i \(0.303498\pi\)
\(192\) 0 0
\(193\) −2.00000 −0.143963 −0.0719816 0.997406i \(-0.522932\pi\)
−0.0719816 + 0.997406i \(0.522932\pi\)
\(194\) 5.25544 0.377318
\(195\) 0 0
\(196\) 1.00000 0.0714286
\(197\) −10.0000 −0.712470 −0.356235 0.934396i \(-0.615940\pi\)
−0.356235 + 0.934396i \(0.615940\pi\)
\(198\) 0 0
\(199\) 14.7446 1.04521 0.522607 0.852574i \(-0.324959\pi\)
0.522607 + 0.852574i \(0.324959\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −8.74456 −0.615265
\(203\) 2.74456 0.192631
\(204\) 0 0
\(205\) −4.00000 −0.279372
\(206\) −10.7446 −0.748609
\(207\) 0 0
\(208\) −4.74456 −0.328976
\(209\) 4.74456 0.328188
\(210\) 0 0
\(211\) −12.0000 −0.826114 −0.413057 0.910705i \(-0.635539\pi\)
−0.413057 + 0.910705i \(0.635539\pi\)
\(212\) 1.25544 0.0862238
\(213\) 0 0
\(214\) −12.0000 −0.820303
\(215\) −4.00000 −0.272798
\(216\) 0 0
\(217\) −6.74456 −0.457851
\(218\) −16.2337 −1.09948
\(219\) 0 0
\(220\) −1.00000 −0.0674200
\(221\) 22.5109 1.51425
\(222\) 0 0
\(223\) 26.7446 1.79095 0.895474 0.445113i \(-0.146837\pi\)
0.895474 + 0.445113i \(0.146837\pi\)
\(224\) −1.00000 −0.0668153
\(225\) 0 0
\(226\) 10.0000 0.665190
\(227\) −20.0000 −1.32745 −0.663723 0.747978i \(-0.731025\pi\)
−0.663723 + 0.747978i \(0.731025\pi\)
\(228\) 0 0
\(229\) −6.00000 −0.396491 −0.198246 0.980152i \(-0.563524\pi\)
−0.198246 + 0.980152i \(0.563524\pi\)
\(230\) −4.74456 −0.312847
\(231\) 0 0
\(232\) −2.74456 −0.180189
\(233\) −20.9783 −1.37433 −0.687165 0.726501i \(-0.741145\pi\)
−0.687165 + 0.726501i \(0.741145\pi\)
\(234\) 0 0
\(235\) −6.74456 −0.439967
\(236\) −2.74456 −0.178656
\(237\) 0 0
\(238\) 4.74456 0.307544
\(239\) 3.25544 0.210577 0.105288 0.994442i \(-0.466423\pi\)
0.105288 + 0.994442i \(0.466423\pi\)
\(240\) 0 0
\(241\) 20.0000 1.28831 0.644157 0.764894i \(-0.277208\pi\)
0.644157 + 0.764894i \(0.277208\pi\)
\(242\) −1.00000 −0.0642824
\(243\) 0 0
\(244\) 12.7446 0.815887
\(245\) −1.00000 −0.0638877
\(246\) 0 0
\(247\) −22.5109 −1.43233
\(248\) 6.74456 0.428280
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) −8.23369 −0.519706 −0.259853 0.965648i \(-0.583674\pi\)
−0.259853 + 0.965648i \(0.583674\pi\)
\(252\) 0 0
\(253\) −4.74456 −0.298288
\(254\) 0 0
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −24.2337 −1.51166 −0.755828 0.654770i \(-0.772765\pi\)
−0.755828 + 0.654770i \(0.772765\pi\)
\(258\) 0 0
\(259\) −10.7446 −0.667635
\(260\) 4.74456 0.294245
\(261\) 0 0
\(262\) 8.74456 0.540241
\(263\) 18.9783 1.17025 0.585125 0.810943i \(-0.301046\pi\)
0.585125 + 0.810943i \(0.301046\pi\)
\(264\) 0 0
\(265\) −1.25544 −0.0771209
\(266\) −4.74456 −0.290908
\(267\) 0 0
\(268\) −4.00000 −0.244339
\(269\) 24.9783 1.52295 0.761475 0.648194i \(-0.224475\pi\)
0.761475 + 0.648194i \(0.224475\pi\)
\(270\) 0 0
\(271\) −30.9783 −1.88179 −0.940897 0.338692i \(-0.890016\pi\)
−0.940897 + 0.338692i \(0.890016\pi\)
\(272\) −4.74456 −0.287681
\(273\) 0 0
\(274\) −19.4891 −1.17738
\(275\) 1.00000 0.0603023
\(276\) 0 0
\(277\) −7.48913 −0.449978 −0.224989 0.974361i \(-0.572235\pi\)
−0.224989 + 0.974361i \(0.572235\pi\)
\(278\) −3.25544 −0.195248
\(279\) 0 0
\(280\) 1.00000 0.0597614
\(281\) −14.0000 −0.835170 −0.417585 0.908638i \(-0.637123\pi\)
−0.417585 + 0.908638i \(0.637123\pi\)
\(282\) 0 0
\(283\) 28.0000 1.66443 0.832214 0.554455i \(-0.187073\pi\)
0.832214 + 0.554455i \(0.187073\pi\)
\(284\) 4.00000 0.237356
\(285\) 0 0
\(286\) 4.74456 0.280552
\(287\) 4.00000 0.236113
\(288\) 0 0
\(289\) 5.51087 0.324169
\(290\) 2.74456 0.161166
\(291\) 0 0
\(292\) −0.744563 −0.0435722
\(293\) −24.7446 −1.44559 −0.722796 0.691061i \(-0.757144\pi\)
−0.722796 + 0.691061i \(0.757144\pi\)
\(294\) 0 0
\(295\) 2.74456 0.159795
\(296\) 10.7446 0.624515
\(297\) 0 0
\(298\) 10.7446 0.622416
\(299\) 22.5109 1.30184
\(300\) 0 0
\(301\) 4.00000 0.230556
\(302\) 20.7446 1.19372
\(303\) 0 0
\(304\) 4.74456 0.272119
\(305\) −12.7446 −0.729752
\(306\) 0 0
\(307\) 21.4891 1.22645 0.613225 0.789909i \(-0.289872\pi\)
0.613225 + 0.789909i \(0.289872\pi\)
\(308\) 1.00000 0.0569803
\(309\) 0 0
\(310\) −6.74456 −0.383065
\(311\) 9.25544 0.524828 0.262414 0.964955i \(-0.415481\pi\)
0.262414 + 0.964955i \(0.415481\pi\)
\(312\) 0 0
\(313\) 32.2337 1.82196 0.910978 0.412455i \(-0.135329\pi\)
0.910978 + 0.412455i \(0.135329\pi\)
\(314\) −23.4891 −1.32557
\(315\) 0 0
\(316\) −4.74456 −0.266903
\(317\) 32.2337 1.81042 0.905212 0.424960i \(-0.139712\pi\)
0.905212 + 0.424960i \(0.139712\pi\)
\(318\) 0 0
\(319\) 2.74456 0.153666
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 4.74456 0.264404
\(323\) −22.5109 −1.25254
\(324\) 0 0
\(325\) −4.74456 −0.263181
\(326\) 4.00000 0.221540
\(327\) 0 0
\(328\) −4.00000 −0.220863
\(329\) 6.74456 0.371840
\(330\) 0 0
\(331\) −30.9783 −1.70272 −0.851359 0.524583i \(-0.824221\pi\)
−0.851359 + 0.524583i \(0.824221\pi\)
\(332\) −8.00000 −0.439057
\(333\) 0 0
\(334\) 0 0
\(335\) 4.00000 0.218543
\(336\) 0 0
\(337\) 26.0000 1.41631 0.708155 0.706057i \(-0.249528\pi\)
0.708155 + 0.706057i \(0.249528\pi\)
\(338\) −9.51087 −0.517323
\(339\) 0 0
\(340\) 4.74456 0.257310
\(341\) −6.74456 −0.365239
\(342\) 0 0
\(343\) 1.00000 0.0539949
\(344\) −4.00000 −0.215666
\(345\) 0 0
\(346\) −14.2337 −0.765208
\(347\) −22.9783 −1.23354 −0.616769 0.787145i \(-0.711559\pi\)
−0.616769 + 0.787145i \(0.711559\pi\)
\(348\) 0 0
\(349\) 19.2554 1.03072 0.515360 0.856974i \(-0.327658\pi\)
0.515360 + 0.856974i \(0.327658\pi\)
\(350\) −1.00000 −0.0534522
\(351\) 0 0
\(352\) −1.00000 −0.0533002
\(353\) 2.74456 0.146078 0.0730392 0.997329i \(-0.476730\pi\)
0.0730392 + 0.997329i \(0.476730\pi\)
\(354\) 0 0
\(355\) −4.00000 −0.212298
\(356\) 7.48913 0.396923
\(357\) 0 0
\(358\) −4.00000 −0.211407
\(359\) 12.7446 0.672632 0.336316 0.941749i \(-0.390819\pi\)
0.336316 + 0.941749i \(0.390819\pi\)
\(360\) 0 0
\(361\) 3.51087 0.184783
\(362\) −3.48913 −0.183384
\(363\) 0 0
\(364\) −4.74456 −0.248683
\(365\) 0.744563 0.0389722
\(366\) 0 0
\(367\) 2.74456 0.143265 0.0716325 0.997431i \(-0.477179\pi\)
0.0716325 + 0.997431i \(0.477179\pi\)
\(368\) −4.74456 −0.247327
\(369\) 0 0
\(370\) −10.7446 −0.558583
\(371\) 1.25544 0.0651791
\(372\) 0 0
\(373\) 28.9783 1.50044 0.750218 0.661190i \(-0.229948\pi\)
0.750218 + 0.661190i \(0.229948\pi\)
\(374\) 4.74456 0.245335
\(375\) 0 0
\(376\) −6.74456 −0.347824
\(377\) −13.0217 −0.670654
\(378\) 0 0
\(379\) −14.5109 −0.745374 −0.372687 0.927957i \(-0.621563\pi\)
−0.372687 + 0.927957i \(0.621563\pi\)
\(380\) −4.74456 −0.243391
\(381\) 0 0
\(382\) −16.0000 −0.818631
\(383\) 37.7228 1.92755 0.963773 0.266724i \(-0.0859413\pi\)
0.963773 + 0.266724i \(0.0859413\pi\)
\(384\) 0 0
\(385\) −1.00000 −0.0509647
\(386\) 2.00000 0.101797
\(387\) 0 0
\(388\) −5.25544 −0.266804
\(389\) −15.4891 −0.785330 −0.392665 0.919682i \(-0.628447\pi\)
−0.392665 + 0.919682i \(0.628447\pi\)
\(390\) 0 0
\(391\) 22.5109 1.13842
\(392\) −1.00000 −0.0505076
\(393\) 0 0
\(394\) 10.0000 0.503793
\(395\) 4.74456 0.238725
\(396\) 0 0
\(397\) 12.5109 0.627903 0.313951 0.949439i \(-0.398347\pi\)
0.313951 + 0.949439i \(0.398347\pi\)
\(398\) −14.7446 −0.739078
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −0.510875 −0.0255119 −0.0127559 0.999919i \(-0.504060\pi\)
−0.0127559 + 0.999919i \(0.504060\pi\)
\(402\) 0 0
\(403\) 32.0000 1.59403
\(404\) 8.74456 0.435058
\(405\) 0 0
\(406\) −2.74456 −0.136210
\(407\) −10.7446 −0.532588
\(408\) 0 0
\(409\) 1.48913 0.0736325 0.0368163 0.999322i \(-0.488278\pi\)
0.0368163 + 0.999322i \(0.488278\pi\)
\(410\) 4.00000 0.197546
\(411\) 0 0
\(412\) 10.7446 0.529347
\(413\) −2.74456 −0.135051
\(414\) 0 0
\(415\) 8.00000 0.392705
\(416\) 4.74456 0.232621
\(417\) 0 0
\(418\) −4.74456 −0.232064
\(419\) −10.7446 −0.524906 −0.262453 0.964945i \(-0.584532\pi\)
−0.262453 + 0.964945i \(0.584532\pi\)
\(420\) 0 0
\(421\) 27.4891 1.33974 0.669869 0.742479i \(-0.266350\pi\)
0.669869 + 0.742479i \(0.266350\pi\)
\(422\) 12.0000 0.584151
\(423\) 0 0
\(424\) −1.25544 −0.0609694
\(425\) −4.74456 −0.230145
\(426\) 0 0
\(427\) 12.7446 0.616753
\(428\) 12.0000 0.580042
\(429\) 0 0
\(430\) 4.00000 0.192897
\(431\) 28.7446 1.38458 0.692288 0.721621i \(-0.256603\pi\)
0.692288 + 0.721621i \(0.256603\pi\)
\(432\) 0 0
\(433\) 25.2554 1.21370 0.606849 0.794817i \(-0.292433\pi\)
0.606849 + 0.794817i \(0.292433\pi\)
\(434\) 6.74456 0.323749
\(435\) 0 0
\(436\) 16.2337 0.777453
\(437\) −22.5109 −1.07684
\(438\) 0 0
\(439\) −30.9783 −1.47851 −0.739256 0.673425i \(-0.764822\pi\)
−0.739256 + 0.673425i \(0.764822\pi\)
\(440\) 1.00000 0.0476731
\(441\) 0 0
\(442\) −22.5109 −1.07073
\(443\) 6.51087 0.309341 0.154670 0.987966i \(-0.450568\pi\)
0.154670 + 0.987966i \(0.450568\pi\)
\(444\) 0 0
\(445\) −7.48913 −0.355019
\(446\) −26.7446 −1.26639
\(447\) 0 0
\(448\) 1.00000 0.0472456
\(449\) −40.9783 −1.93388 −0.966942 0.254998i \(-0.917925\pi\)
−0.966942 + 0.254998i \(0.917925\pi\)
\(450\) 0 0
\(451\) 4.00000 0.188353
\(452\) −10.0000 −0.470360
\(453\) 0 0
\(454\) 20.0000 0.938647
\(455\) 4.74456 0.222429
\(456\) 0 0
\(457\) −19.4891 −0.911663 −0.455831 0.890066i \(-0.650658\pi\)
−0.455831 + 0.890066i \(0.650658\pi\)
\(458\) 6.00000 0.280362
\(459\) 0 0
\(460\) 4.74456 0.221216
\(461\) 23.7228 1.10488 0.552441 0.833552i \(-0.313697\pi\)
0.552441 + 0.833552i \(0.313697\pi\)
\(462\) 0 0
\(463\) 12.7446 0.592290 0.296145 0.955143i \(-0.404299\pi\)
0.296145 + 0.955143i \(0.404299\pi\)
\(464\) 2.74456 0.127413
\(465\) 0 0
\(466\) 20.9783 0.971799
\(467\) 28.9783 1.34095 0.670477 0.741931i \(-0.266090\pi\)
0.670477 + 0.741931i \(0.266090\pi\)
\(468\) 0 0
\(469\) −4.00000 −0.184703
\(470\) 6.74456 0.311103
\(471\) 0 0
\(472\) 2.74456 0.126329
\(473\) 4.00000 0.183920
\(474\) 0 0
\(475\) 4.74456 0.217695
\(476\) −4.74456 −0.217467
\(477\) 0 0
\(478\) −3.25544 −0.148900
\(479\) 18.5109 0.845783 0.422892 0.906180i \(-0.361015\pi\)
0.422892 + 0.906180i \(0.361015\pi\)
\(480\) 0 0
\(481\) 50.9783 2.32441
\(482\) −20.0000 −0.910975
\(483\) 0 0
\(484\) 1.00000 0.0454545
\(485\) 5.25544 0.238637
\(486\) 0 0
\(487\) 20.7446 0.940026 0.470013 0.882660i \(-0.344249\pi\)
0.470013 + 0.882660i \(0.344249\pi\)
\(488\) −12.7446 −0.576919
\(489\) 0 0
\(490\) 1.00000 0.0451754
\(491\) 14.9783 0.675959 0.337979 0.941153i \(-0.390257\pi\)
0.337979 + 0.941153i \(0.390257\pi\)
\(492\) 0 0
\(493\) −13.0217 −0.586470
\(494\) 22.5109 1.01281
\(495\) 0 0
\(496\) −6.74456 −0.302840
\(497\) 4.00000 0.179425
\(498\) 0 0
\(499\) 9.48913 0.424792 0.212396 0.977184i \(-0.431873\pi\)
0.212396 + 0.977184i \(0.431873\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) 8.23369 0.367487
\(503\) −29.4891 −1.31486 −0.657428 0.753518i \(-0.728355\pi\)
−0.657428 + 0.753518i \(0.728355\pi\)
\(504\) 0 0
\(505\) −8.74456 −0.389128
\(506\) 4.74456 0.210922
\(507\) 0 0
\(508\) 0 0
\(509\) 12.5109 0.554535 0.277267 0.960793i \(-0.410571\pi\)
0.277267 + 0.960793i \(0.410571\pi\)
\(510\) 0 0
\(511\) −0.744563 −0.0329375
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) 24.2337 1.06890
\(515\) −10.7446 −0.473462
\(516\) 0 0
\(517\) 6.74456 0.296626
\(518\) 10.7446 0.472089
\(519\) 0 0
\(520\) −4.74456 −0.208063
\(521\) 2.00000 0.0876216 0.0438108 0.999040i \(-0.486050\pi\)
0.0438108 + 0.999040i \(0.486050\pi\)
\(522\) 0 0
\(523\) 5.48913 0.240023 0.120011 0.992773i \(-0.461707\pi\)
0.120011 + 0.992773i \(0.461707\pi\)
\(524\) −8.74456 −0.382008
\(525\) 0 0
\(526\) −18.9783 −0.827491
\(527\) 32.0000 1.39394
\(528\) 0 0
\(529\) −0.489125 −0.0212663
\(530\) 1.25544 0.0545327
\(531\) 0 0
\(532\) 4.74456 0.205703
\(533\) −18.9783 −0.822039
\(534\) 0 0
\(535\) −12.0000 −0.518805
\(536\) 4.00000 0.172774
\(537\) 0 0
\(538\) −24.9783 −1.07689
\(539\) 1.00000 0.0430730
\(540\) 0 0
\(541\) 36.2337 1.55781 0.778904 0.627143i \(-0.215776\pi\)
0.778904 + 0.627143i \(0.215776\pi\)
\(542\) 30.9783 1.33063
\(543\) 0 0
\(544\) 4.74456 0.203421
\(545\) −16.2337 −0.695375
\(546\) 0 0
\(547\) −30.9783 −1.32453 −0.662267 0.749268i \(-0.730406\pi\)
−0.662267 + 0.749268i \(0.730406\pi\)
\(548\) 19.4891 0.832534
\(549\) 0 0
\(550\) −1.00000 −0.0426401
\(551\) 13.0217 0.554745
\(552\) 0 0
\(553\) −4.74456 −0.201759
\(554\) 7.48913 0.318182
\(555\) 0 0
\(556\) 3.25544 0.138061
\(557\) −44.9783 −1.90579 −0.952895 0.303301i \(-0.901911\pi\)
−0.952895 + 0.303301i \(0.901911\pi\)
\(558\) 0 0
\(559\) −18.9783 −0.802694
\(560\) −1.00000 −0.0422577
\(561\) 0 0
\(562\) 14.0000 0.590554
\(563\) 17.4891 0.737079 0.368539 0.929612i \(-0.379858\pi\)
0.368539 + 0.929612i \(0.379858\pi\)
\(564\) 0 0
\(565\) 10.0000 0.420703
\(566\) −28.0000 −1.17693
\(567\) 0 0
\(568\) −4.00000 −0.167836
\(569\) −39.4891 −1.65547 −0.827735 0.561119i \(-0.810371\pi\)
−0.827735 + 0.561119i \(0.810371\pi\)
\(570\) 0 0
\(571\) 5.48913 0.229713 0.114856 0.993382i \(-0.463359\pi\)
0.114856 + 0.993382i \(0.463359\pi\)
\(572\) −4.74456 −0.198380
\(573\) 0 0
\(574\) −4.00000 −0.166957
\(575\) −4.74456 −0.197862
\(576\) 0 0
\(577\) 2.74456 0.114258 0.0571288 0.998367i \(-0.481805\pi\)
0.0571288 + 0.998367i \(0.481805\pi\)
\(578\) −5.51087 −0.229222
\(579\) 0 0
\(580\) −2.74456 −0.113962
\(581\) −8.00000 −0.331896
\(582\) 0 0
\(583\) 1.25544 0.0519949
\(584\) 0.744563 0.0308102
\(585\) 0 0
\(586\) 24.7446 1.02219
\(587\) −40.9783 −1.69135 −0.845677 0.533696i \(-0.820803\pi\)
−0.845677 + 0.533696i \(0.820803\pi\)
\(588\) 0 0
\(589\) −32.0000 −1.31854
\(590\) −2.74456 −0.112992
\(591\) 0 0
\(592\) −10.7446 −0.441599
\(593\) 5.76631 0.236794 0.118397 0.992966i \(-0.462224\pi\)
0.118397 + 0.992966i \(0.462224\pi\)
\(594\) 0 0
\(595\) 4.74456 0.194508
\(596\) −10.7446 −0.440114
\(597\) 0 0
\(598\) −22.5109 −0.920538
\(599\) 12.0000 0.490307 0.245153 0.969484i \(-0.421162\pi\)
0.245153 + 0.969484i \(0.421162\pi\)
\(600\) 0 0
\(601\) −10.5109 −0.428748 −0.214374 0.976752i \(-0.568771\pi\)
−0.214374 + 0.976752i \(0.568771\pi\)
\(602\) −4.00000 −0.163028
\(603\) 0 0
\(604\) −20.7446 −0.844084
\(605\) −1.00000 −0.0406558
\(606\) 0 0
\(607\) 8.00000 0.324710 0.162355 0.986732i \(-0.448091\pi\)
0.162355 + 0.986732i \(0.448091\pi\)
\(608\) −4.74456 −0.192417
\(609\) 0 0
\(610\) 12.7446 0.516012
\(611\) −32.0000 −1.29458
\(612\) 0 0
\(613\) −11.4891 −0.464041 −0.232021 0.972711i \(-0.574534\pi\)
−0.232021 + 0.972711i \(0.574534\pi\)
\(614\) −21.4891 −0.867231
\(615\) 0 0
\(616\) −1.00000 −0.0402911
\(617\) 10.0000 0.402585 0.201292 0.979531i \(-0.435486\pi\)
0.201292 + 0.979531i \(0.435486\pi\)
\(618\) 0 0
\(619\) −10.7446 −0.431860 −0.215930 0.976409i \(-0.569278\pi\)
−0.215930 + 0.976409i \(0.569278\pi\)
\(620\) 6.74456 0.270868
\(621\) 0 0
\(622\) −9.25544 −0.371109
\(623\) 7.48913 0.300045
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −32.2337 −1.28832
\(627\) 0 0
\(628\) 23.4891 0.937318
\(629\) 50.9783 2.03264
\(630\) 0 0
\(631\) 42.9783 1.71094 0.855469 0.517855i \(-0.173269\pi\)
0.855469 + 0.517855i \(0.173269\pi\)
\(632\) 4.74456 0.188729
\(633\) 0 0
\(634\) −32.2337 −1.28016
\(635\) 0 0
\(636\) 0 0
\(637\) −4.74456 −0.187986
\(638\) −2.74456 −0.108658
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 11.4891 0.453793 0.226897 0.973919i \(-0.427142\pi\)
0.226897 + 0.973919i \(0.427142\pi\)
\(642\) 0 0
\(643\) −22.4674 −0.886027 −0.443013 0.896515i \(-0.646091\pi\)
−0.443013 + 0.896515i \(0.646091\pi\)
\(644\) −4.74456 −0.186962
\(645\) 0 0
\(646\) 22.5109 0.885679
\(647\) 2.74456 0.107900 0.0539499 0.998544i \(-0.482819\pi\)
0.0539499 + 0.998544i \(0.482819\pi\)
\(648\) 0 0
\(649\) −2.74456 −0.107734
\(650\) 4.74456 0.186097
\(651\) 0 0
\(652\) −4.00000 −0.156652
\(653\) 41.7228 1.63274 0.816370 0.577529i \(-0.195983\pi\)
0.816370 + 0.577529i \(0.195983\pi\)
\(654\) 0 0
\(655\) 8.74456 0.341678
\(656\) 4.00000 0.156174
\(657\) 0 0
\(658\) −6.74456 −0.262930
\(659\) 18.5109 0.721081 0.360541 0.932744i \(-0.382592\pi\)
0.360541 + 0.932744i \(0.382592\pi\)
\(660\) 0 0
\(661\) 22.0000 0.855701 0.427850 0.903850i \(-0.359271\pi\)
0.427850 + 0.903850i \(0.359271\pi\)
\(662\) 30.9783 1.20400
\(663\) 0 0
\(664\) 8.00000 0.310460
\(665\) −4.74456 −0.183986
\(666\) 0 0
\(667\) −13.0217 −0.504204
\(668\) 0 0
\(669\) 0 0
\(670\) −4.00000 −0.154533
\(671\) 12.7446 0.491998
\(672\) 0 0
\(673\) −24.9783 −0.962841 −0.481420 0.876490i \(-0.659879\pi\)
−0.481420 + 0.876490i \(0.659879\pi\)
\(674\) −26.0000 −1.00148
\(675\) 0 0
\(676\) 9.51087 0.365803
\(677\) −1.76631 −0.0678849 −0.0339424 0.999424i \(-0.510806\pi\)
−0.0339424 + 0.999424i \(0.510806\pi\)
\(678\) 0 0
\(679\) −5.25544 −0.201685
\(680\) −4.74456 −0.181946
\(681\) 0 0
\(682\) 6.74456 0.258263
\(683\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(684\) 0 0
\(685\) −19.4891 −0.744641
\(686\) −1.00000 −0.0381802
\(687\) 0 0
\(688\) 4.00000 0.152499
\(689\) −5.95650 −0.226925
\(690\) 0 0
\(691\) 36.2337 1.37839 0.689197 0.724574i \(-0.257963\pi\)
0.689197 + 0.724574i \(0.257963\pi\)
\(692\) 14.2337 0.541084
\(693\) 0 0
\(694\) 22.9783 0.872242
\(695\) −3.25544 −0.123486
\(696\) 0 0
\(697\) −18.9783 −0.718853
\(698\) −19.2554 −0.728829
\(699\) 0 0
\(700\) 1.00000 0.0377964
\(701\) 12.2337 0.462060 0.231030 0.972947i \(-0.425790\pi\)
0.231030 + 0.972947i \(0.425790\pi\)
\(702\) 0 0
\(703\) −50.9783 −1.92268
\(704\) 1.00000 0.0376889
\(705\) 0 0
\(706\) −2.74456 −0.103293
\(707\) 8.74456 0.328873
\(708\) 0 0
\(709\) 23.4891 0.882153 0.441076 0.897470i \(-0.354597\pi\)
0.441076 + 0.897470i \(0.354597\pi\)
\(710\) 4.00000 0.150117
\(711\) 0 0
\(712\) −7.48913 −0.280667
\(713\) 32.0000 1.19841
\(714\) 0 0
\(715\) 4.74456 0.177437
\(716\) 4.00000 0.149487
\(717\) 0 0
\(718\) −12.7446 −0.475623
\(719\) 49.7228 1.85435 0.927174 0.374631i \(-0.122231\pi\)
0.927174 + 0.374631i \(0.122231\pi\)
\(720\) 0 0
\(721\) 10.7446 0.400148
\(722\) −3.51087 −0.130661
\(723\) 0 0
\(724\) 3.48913 0.129672
\(725\) 2.74456 0.101930
\(726\) 0 0
\(727\) −20.2337 −0.750426 −0.375213 0.926939i \(-0.622430\pi\)
−0.375213 + 0.926939i \(0.622430\pi\)
\(728\) 4.74456 0.175845
\(729\) 0 0
\(730\) −0.744563 −0.0275575
\(731\) −18.9783 −0.701936
\(732\) 0 0
\(733\) 18.2337 0.673477 0.336738 0.941598i \(-0.390676\pi\)
0.336738 + 0.941598i \(0.390676\pi\)
\(734\) −2.74456 −0.101304
\(735\) 0 0
\(736\) 4.74456 0.174887
\(737\) −4.00000 −0.147342
\(738\) 0 0
\(739\) 14.9783 0.550984 0.275492 0.961303i \(-0.411159\pi\)
0.275492 + 0.961303i \(0.411159\pi\)
\(740\) 10.7446 0.394978
\(741\) 0 0
\(742\) −1.25544 −0.0460886
\(743\) 18.9783 0.696244 0.348122 0.937449i \(-0.386819\pi\)
0.348122 + 0.937449i \(0.386819\pi\)
\(744\) 0 0
\(745\) 10.7446 0.393650
\(746\) −28.9783 −1.06097
\(747\) 0 0
\(748\) −4.74456 −0.173478
\(749\) 12.0000 0.438470
\(750\) 0 0
\(751\) 20.0000 0.729810 0.364905 0.931045i \(-0.381101\pi\)
0.364905 + 0.931045i \(0.381101\pi\)
\(752\) 6.74456 0.245949
\(753\) 0 0
\(754\) 13.0217 0.474224
\(755\) 20.7446 0.754972
\(756\) 0 0
\(757\) −30.7446 −1.11743 −0.558715 0.829360i \(-0.688705\pi\)
−0.558715 + 0.829360i \(0.688705\pi\)
\(758\) 14.5109 0.527059
\(759\) 0 0
\(760\) 4.74456 0.172103
\(761\) 6.51087 0.236019 0.118010 0.993012i \(-0.462349\pi\)
0.118010 + 0.993012i \(0.462349\pi\)
\(762\) 0 0
\(763\) 16.2337 0.587699
\(764\) 16.0000 0.578860
\(765\) 0 0
\(766\) −37.7228 −1.36298
\(767\) 13.0217 0.470188
\(768\) 0 0
\(769\) −8.00000 −0.288487 −0.144244 0.989542i \(-0.546075\pi\)
−0.144244 + 0.989542i \(0.546075\pi\)
\(770\) 1.00000 0.0360375
\(771\) 0 0
\(772\) −2.00000 −0.0719816
\(773\) 28.5109 1.02546 0.512732 0.858548i \(-0.328633\pi\)
0.512732 + 0.858548i \(0.328633\pi\)
\(774\) 0 0
\(775\) −6.74456 −0.242272
\(776\) 5.25544 0.188659
\(777\) 0 0
\(778\) 15.4891 0.555312
\(779\) 18.9783 0.679966
\(780\) 0 0
\(781\) 4.00000 0.143131
\(782\) −22.5109 −0.804987
\(783\) 0 0
\(784\) 1.00000 0.0357143
\(785\) −23.4891 −0.838363
\(786\) 0 0
\(787\) −17.4891 −0.623420 −0.311710 0.950177i \(-0.600902\pi\)
−0.311710 + 0.950177i \(0.600902\pi\)
\(788\) −10.0000 −0.356235
\(789\) 0 0
\(790\) −4.74456 −0.168804
\(791\) −10.0000 −0.355559
\(792\) 0 0
\(793\) −60.4674 −2.14726
\(794\) −12.5109 −0.443994
\(795\) 0 0
\(796\) 14.7446 0.522607
\(797\) −26.4674 −0.937523 −0.468761 0.883325i \(-0.655300\pi\)
−0.468761 + 0.883325i \(0.655300\pi\)
\(798\) 0 0
\(799\) −32.0000 −1.13208
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 0.510875 0.0180396
\(803\) −0.744563 −0.0262750
\(804\) 0 0
\(805\) 4.74456 0.167224
\(806\) −32.0000 −1.12715
\(807\) 0 0
\(808\) −8.74456 −0.307633
\(809\) 34.4674 1.21181 0.605904 0.795538i \(-0.292811\pi\)
0.605904 + 0.795538i \(0.292811\pi\)
\(810\) 0 0
\(811\) −7.25544 −0.254773 −0.127386 0.991853i \(-0.540659\pi\)
−0.127386 + 0.991853i \(0.540659\pi\)
\(812\) 2.74456 0.0963153
\(813\) 0 0
\(814\) 10.7446 0.376597
\(815\) 4.00000 0.140114
\(816\) 0 0
\(817\) 18.9783 0.663965
\(818\) −1.48913 −0.0520660
\(819\) 0 0
\(820\) −4.00000 −0.139686
\(821\) 49.7228 1.73534 0.867669 0.497142i \(-0.165617\pi\)
0.867669 + 0.497142i \(0.165617\pi\)
\(822\) 0 0
\(823\) 42.2337 1.47217 0.736087 0.676887i \(-0.236671\pi\)
0.736087 + 0.676887i \(0.236671\pi\)
\(824\) −10.7446 −0.374305
\(825\) 0 0
\(826\) 2.74456 0.0954955
\(827\) −4.00000 −0.139094 −0.0695468 0.997579i \(-0.522155\pi\)
−0.0695468 + 0.997579i \(0.522155\pi\)
\(828\) 0 0
\(829\) −54.4674 −1.89173 −0.945865 0.324560i \(-0.894784\pi\)
−0.945865 + 0.324560i \(0.894784\pi\)
\(830\) −8.00000 −0.277684
\(831\) 0 0
\(832\) −4.74456 −0.164488
\(833\) −4.74456 −0.164389
\(834\) 0 0
\(835\) 0 0
\(836\) 4.74456 0.164094
\(837\) 0 0
\(838\) 10.7446 0.371165
\(839\) 14.7446 0.509039 0.254519 0.967068i \(-0.418083\pi\)
0.254519 + 0.967068i \(0.418083\pi\)
\(840\) 0 0
\(841\) −21.4674 −0.740254
\(842\) −27.4891 −0.947338
\(843\) 0 0
\(844\) −12.0000 −0.413057
\(845\) −9.51087 −0.327184
\(846\) 0 0
\(847\) 1.00000 0.0343604
\(848\) 1.25544 0.0431119
\(849\) 0 0
\(850\) 4.74456 0.162737
\(851\) 50.9783 1.74751
\(852\) 0 0
\(853\) −11.2554 −0.385379 −0.192689 0.981260i \(-0.561721\pi\)
−0.192689 + 0.981260i \(0.561721\pi\)
\(854\) −12.7446 −0.436110
\(855\) 0 0
\(856\) −12.0000 −0.410152
\(857\) −37.2119 −1.27114 −0.635568 0.772045i \(-0.719234\pi\)
−0.635568 + 0.772045i \(0.719234\pi\)
\(858\) 0 0
\(859\) 51.2119 1.74733 0.873664 0.486529i \(-0.161737\pi\)
0.873664 + 0.486529i \(0.161737\pi\)
\(860\) −4.00000 −0.136399
\(861\) 0 0
\(862\) −28.7446 −0.979044
\(863\) 5.76631 0.196288 0.0981438 0.995172i \(-0.468709\pi\)
0.0981438 + 0.995172i \(0.468709\pi\)
\(864\) 0 0
\(865\) −14.2337 −0.483960
\(866\) −25.2554 −0.858215
\(867\) 0 0
\(868\) −6.74456 −0.228925
\(869\) −4.74456 −0.160948
\(870\) 0 0
\(871\) 18.9783 0.643053
\(872\) −16.2337 −0.549742
\(873\) 0 0
\(874\) 22.5109 0.761442
\(875\) −1.00000 −0.0338062
\(876\) 0 0
\(877\) 42.4674 1.43402 0.717011 0.697062i \(-0.245510\pi\)
0.717011 + 0.697062i \(0.245510\pi\)
\(878\) 30.9783 1.04547
\(879\) 0 0
\(880\) −1.00000 −0.0337100
\(881\) 32.5109 1.09532 0.547660 0.836701i \(-0.315519\pi\)
0.547660 + 0.836701i \(0.315519\pi\)
\(882\) 0 0
\(883\) −8.00000 −0.269221 −0.134611 0.990899i \(-0.542978\pi\)
−0.134611 + 0.990899i \(0.542978\pi\)
\(884\) 22.5109 0.757123
\(885\) 0 0
\(886\) −6.51087 −0.218737
\(887\) 18.5109 0.621534 0.310767 0.950486i \(-0.399414\pi\)
0.310767 + 0.950486i \(0.399414\pi\)
\(888\) 0 0
\(889\) 0 0
\(890\) 7.48913 0.251036
\(891\) 0 0
\(892\) 26.7446 0.895474
\(893\) 32.0000 1.07084
\(894\) 0 0
\(895\) −4.00000 −0.133705
\(896\) −1.00000 −0.0334077
\(897\) 0 0
\(898\) 40.9783 1.36746
\(899\) −18.5109 −0.617372
\(900\) 0 0
\(901\) −5.95650 −0.198440
\(902\) −4.00000 −0.133185
\(903\) 0 0
\(904\) 10.0000 0.332595
\(905\) −3.48913 −0.115982
\(906\) 0 0
\(907\) −37.4891 −1.24481 −0.622403 0.782697i \(-0.713843\pi\)
−0.622403 + 0.782697i \(0.713843\pi\)
\(908\) −20.0000 −0.663723
\(909\) 0 0
\(910\) −4.74456 −0.157281
\(911\) −20.0000 −0.662630 −0.331315 0.943520i \(-0.607492\pi\)
−0.331315 + 0.943520i \(0.607492\pi\)
\(912\) 0 0
\(913\) −8.00000 −0.264761
\(914\) 19.4891 0.644643
\(915\) 0 0
\(916\) −6.00000 −0.198246
\(917\) −8.74456 −0.288771
\(918\) 0 0
\(919\) 25.2119 0.831665 0.415833 0.909441i \(-0.363490\pi\)
0.415833 + 0.909441i \(0.363490\pi\)
\(920\) −4.74456 −0.156424
\(921\) 0 0
\(922\) −23.7228 −0.781269
\(923\) −18.9783 −0.624677
\(924\) 0 0
\(925\) −10.7446 −0.353279
\(926\) −12.7446 −0.418812
\(927\) 0 0
\(928\) −2.74456 −0.0900947
\(929\) −16.9783 −0.557038 −0.278519 0.960431i \(-0.589844\pi\)
−0.278519 + 0.960431i \(0.589844\pi\)
\(930\) 0 0
\(931\) 4.74456 0.155497
\(932\) −20.9783 −0.687165
\(933\) 0 0
\(934\) −28.9783 −0.948197
\(935\) 4.74456 0.155164
\(936\) 0 0
\(937\) −7.25544 −0.237025 −0.118512 0.992953i \(-0.537813\pi\)
−0.118512 + 0.992953i \(0.537813\pi\)
\(938\) 4.00000 0.130605
\(939\) 0 0
\(940\) −6.74456 −0.219983
\(941\) 22.2337 0.724798 0.362399 0.932023i \(-0.381958\pi\)
0.362399 + 0.932023i \(0.381958\pi\)
\(942\) 0 0
\(943\) −18.9783 −0.618017
\(944\) −2.74456 −0.0893279
\(945\) 0 0
\(946\) −4.00000 −0.130051
\(947\) 8.00000 0.259965 0.129983 0.991516i \(-0.458508\pi\)
0.129983 + 0.991516i \(0.458508\pi\)
\(948\) 0 0
\(949\) 3.53262 0.114674
\(950\) −4.74456 −0.153934
\(951\) 0 0
\(952\) 4.74456 0.153772
\(953\) −6.00000 −0.194359 −0.0971795 0.995267i \(-0.530982\pi\)
−0.0971795 + 0.995267i \(0.530982\pi\)
\(954\) 0 0
\(955\) −16.0000 −0.517748
\(956\) 3.25544 0.105288
\(957\) 0 0
\(958\) −18.5109 −0.598059
\(959\) 19.4891 0.629337
\(960\) 0 0
\(961\) 14.4891 0.467391
\(962\) −50.9783 −1.64360
\(963\) 0 0
\(964\) 20.0000 0.644157
\(965\) 2.00000 0.0643823
\(966\) 0 0
\(967\) −5.02175 −0.161489 −0.0807443 0.996735i \(-0.525730\pi\)
−0.0807443 + 0.996735i \(0.525730\pi\)
\(968\) −1.00000 −0.0321412
\(969\) 0 0
\(970\) −5.25544 −0.168742
\(971\) 37.7228 1.21058 0.605291 0.796004i \(-0.293057\pi\)
0.605291 + 0.796004i \(0.293057\pi\)
\(972\) 0 0
\(973\) 3.25544 0.104365
\(974\) −20.7446 −0.664699
\(975\) 0 0
\(976\) 12.7446 0.407944
\(977\) 32.9783 1.05507 0.527534 0.849534i \(-0.323117\pi\)
0.527534 + 0.849534i \(0.323117\pi\)
\(978\) 0 0
\(979\) 7.48913 0.239353
\(980\) −1.00000 −0.0319438
\(981\) 0 0
\(982\) −14.9783 −0.477975
\(983\) 32.2337 1.02809 0.514047 0.857762i \(-0.328146\pi\)
0.514047 + 0.857762i \(0.328146\pi\)
\(984\) 0 0
\(985\) 10.0000 0.318626
\(986\) 13.0217 0.414697
\(987\) 0 0
\(988\) −22.5109 −0.716166
\(989\) −18.9783 −0.603473
\(990\) 0 0
\(991\) −21.4891 −0.682625 −0.341312 0.939950i \(-0.610871\pi\)
−0.341312 + 0.939950i \(0.610871\pi\)
\(992\) 6.74456 0.214140
\(993\) 0 0
\(994\) −4.00000 −0.126872
\(995\) −14.7446 −0.467434
\(996\) 0 0
\(997\) −41.2119 −1.30520 −0.652598 0.757705i \(-0.726321\pi\)
−0.652598 + 0.757705i \(0.726321\pi\)
\(998\) −9.48913 −0.300373
\(999\) 0 0
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 6930.2.a.bo.1.1 2
3.2 odd 2 770.2.a.k.1.1 2
12.11 even 2 6160.2.a.r.1.1 2
15.2 even 4 3850.2.c.y.1849.4 4
15.8 even 4 3850.2.c.y.1849.1 4
15.14 odd 2 3850.2.a.bc.1.2 2
21.20 even 2 5390.2.a.bq.1.2 2
33.32 even 2 8470.2.a.bu.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
770.2.a.k.1.1 2 3.2 odd 2
3850.2.a.bc.1.2 2 15.14 odd 2
3850.2.c.y.1849.1 4 15.8 even 4
3850.2.c.y.1849.4 4 15.2 even 4
5390.2.a.bq.1.2 2 21.20 even 2
6160.2.a.r.1.1 2 12.11 even 2
6930.2.a.bo.1.1 2 1.1 even 1 trivial
8470.2.a.bu.1.2 2 33.32 even 2