Properties

Label 585.2.a.k.1.1
Level $585$
Weight $2$
Character 585.1
Self dual yes
Analytic conductor $4.671$
Analytic rank $0$
Dimension $2$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 585.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(4.67124851824\)
Analytic rank: \(0\)
Dimension: \(2\)
Coefficient field: \(\Q(\zeta_{12})^+\)
Defining polynomial: \( x^{2} - 3 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 65)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-1.73205\) of defining polynomial
Character \(\chi\) \(=\) 585.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} +O(q^{10})\) \(q-1.73205 q^{2} +1.00000 q^{4} +1.00000 q^{5} +2.00000 q^{7} +1.73205 q^{8} -1.73205 q^{10} +1.26795 q^{11} +1.00000 q^{13} -3.46410 q^{14} -5.00000 q^{16} -3.46410 q^{17} +4.19615 q^{19} +1.00000 q^{20} -2.19615 q^{22} -4.73205 q^{23} +1.00000 q^{25} -1.73205 q^{26} +2.00000 q^{28} +9.46410 q^{29} -0.196152 q^{31} +5.19615 q^{32} +6.00000 q^{34} +2.00000 q^{35} -4.00000 q^{37} -7.26795 q^{38} +1.73205 q^{40} +3.46410 q^{41} +10.1962 q^{43} +1.26795 q^{44} +8.19615 q^{46} -6.00000 q^{47} -3.00000 q^{49} -1.73205 q^{50} +1.00000 q^{52} +10.3923 q^{53} +1.26795 q^{55} +3.46410 q^{56} -16.3923 q^{58} +15.1244 q^{59} +12.3923 q^{61} +0.339746 q^{62} +1.00000 q^{64} +1.00000 q^{65} -14.3923 q^{67} -3.46410 q^{68} -3.46410 q^{70} -1.26795 q^{71} -4.00000 q^{73} +6.92820 q^{74} +4.19615 q^{76} +2.53590 q^{77} +12.3923 q^{79} -5.00000 q^{80} -6.00000 q^{82} +6.00000 q^{83} -3.46410 q^{85} -17.6603 q^{86} +2.19615 q^{88} -0.928203 q^{89} +2.00000 q^{91} -4.73205 q^{92} +10.3923 q^{94} +4.19615 q^{95} +2.00000 q^{97} +5.19615 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7}+O(q^{10}) \) Copy content Toggle raw display \( 2 q + 2 q^{4} + 2 q^{5} + 4 q^{7} + 6 q^{11} + 2 q^{13} - 10 q^{16} - 2 q^{19} + 2 q^{20} + 6 q^{22} - 6 q^{23} + 2 q^{25} + 4 q^{28} + 12 q^{29} + 10 q^{31} + 12 q^{34} + 4 q^{35} - 8 q^{37} - 18 q^{38} + 10 q^{43} + 6 q^{44} + 6 q^{46} - 12 q^{47} - 6 q^{49} + 2 q^{52} + 6 q^{55} - 12 q^{58} + 6 q^{59} + 4 q^{61} + 18 q^{62} + 2 q^{64} + 2 q^{65} - 8 q^{67} - 6 q^{71} - 8 q^{73} - 2 q^{76} + 12 q^{77} + 4 q^{79} - 10 q^{80} - 12 q^{82} + 12 q^{83} - 18 q^{86} - 6 q^{88} + 12 q^{89} + 4 q^{91} - 6 q^{92} - 2 q^{95} + 4 q^{97}+O(q^{100}) \) 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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.73205 −1.22474 −0.612372 0.790569i \(-0.709785\pi\)
−0.612372 + 0.790569i \(0.709785\pi\)
\(3\) 0 0
\(4\) 1.00000 0.500000
\(5\) 1.00000 0.447214
\(6\) 0 0
\(7\) 2.00000 0.755929 0.377964 0.925820i \(-0.376624\pi\)
0.377964 + 0.925820i \(0.376624\pi\)
\(8\) 1.73205 0.612372
\(9\) 0 0
\(10\) −1.73205 −0.547723
\(11\) 1.26795 0.382301 0.191151 0.981561i \(-0.438778\pi\)
0.191151 + 0.981561i \(0.438778\pi\)
\(12\) 0 0
\(13\) 1.00000 0.277350
\(14\) −3.46410 −0.925820
\(15\) 0 0
\(16\) −5.00000 −1.25000
\(17\) −3.46410 −0.840168 −0.420084 0.907485i \(-0.637999\pi\)
−0.420084 + 0.907485i \(0.637999\pi\)
\(18\) 0 0
\(19\) 4.19615 0.962663 0.481332 0.876539i \(-0.340153\pi\)
0.481332 + 0.876539i \(0.340153\pi\)
\(20\) 1.00000 0.223607
\(21\) 0 0
\(22\) −2.19615 −0.468221
\(23\) −4.73205 −0.986701 −0.493350 0.869831i \(-0.664228\pi\)
−0.493350 + 0.869831i \(0.664228\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) −1.73205 −0.339683
\(27\) 0 0
\(28\) 2.00000 0.377964
\(29\) 9.46410 1.75744 0.878720 0.477338i \(-0.158398\pi\)
0.878720 + 0.477338i \(0.158398\pi\)
\(30\) 0 0
\(31\) −0.196152 −0.0352300 −0.0176150 0.999845i \(-0.505607\pi\)
−0.0176150 + 0.999845i \(0.505607\pi\)
\(32\) 5.19615 0.918559
\(33\) 0 0
\(34\) 6.00000 1.02899
\(35\) 2.00000 0.338062
\(36\) 0 0
\(37\) −4.00000 −0.657596 −0.328798 0.944400i \(-0.606644\pi\)
−0.328798 + 0.944400i \(0.606644\pi\)
\(38\) −7.26795 −1.17902
\(39\) 0 0
\(40\) 1.73205 0.273861
\(41\) 3.46410 0.541002 0.270501 0.962720i \(-0.412811\pi\)
0.270501 + 0.962720i \(0.412811\pi\)
\(42\) 0 0
\(43\) 10.1962 1.55490 0.777449 0.628946i \(-0.216513\pi\)
0.777449 + 0.628946i \(0.216513\pi\)
\(44\) 1.26795 0.191151
\(45\) 0 0
\(46\) 8.19615 1.20846
\(47\) −6.00000 −0.875190 −0.437595 0.899172i \(-0.644170\pi\)
−0.437595 + 0.899172i \(0.644170\pi\)
\(48\) 0 0
\(49\) −3.00000 −0.428571
\(50\) −1.73205 −0.244949
\(51\) 0 0
\(52\) 1.00000 0.138675
\(53\) 10.3923 1.42749 0.713746 0.700404i \(-0.246997\pi\)
0.713746 + 0.700404i \(0.246997\pi\)
\(54\) 0 0
\(55\) 1.26795 0.170970
\(56\) 3.46410 0.462910
\(57\) 0 0
\(58\) −16.3923 −2.15242
\(59\) 15.1244 1.96902 0.984512 0.175319i \(-0.0560957\pi\)
0.984512 + 0.175319i \(0.0560957\pi\)
\(60\) 0 0
\(61\) 12.3923 1.58667 0.793336 0.608784i \(-0.208342\pi\)
0.793336 + 0.608784i \(0.208342\pi\)
\(62\) 0.339746 0.0431478
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 1.00000 0.124035
\(66\) 0 0
\(67\) −14.3923 −1.75830 −0.879150 0.476545i \(-0.841889\pi\)
−0.879150 + 0.476545i \(0.841889\pi\)
\(68\) −3.46410 −0.420084
\(69\) 0 0
\(70\) −3.46410 −0.414039
\(71\) −1.26795 −0.150478 −0.0752389 0.997166i \(-0.523972\pi\)
−0.0752389 + 0.997166i \(0.523972\pi\)
\(72\) 0 0
\(73\) −4.00000 −0.468165 −0.234082 0.972217i \(-0.575209\pi\)
−0.234082 + 0.972217i \(0.575209\pi\)
\(74\) 6.92820 0.805387
\(75\) 0 0
\(76\) 4.19615 0.481332
\(77\) 2.53590 0.288992
\(78\) 0 0
\(79\) 12.3923 1.39424 0.697122 0.716953i \(-0.254464\pi\)
0.697122 + 0.716953i \(0.254464\pi\)
\(80\) −5.00000 −0.559017
\(81\) 0 0
\(82\) −6.00000 −0.662589
\(83\) 6.00000 0.658586 0.329293 0.944228i \(-0.393190\pi\)
0.329293 + 0.944228i \(0.393190\pi\)
\(84\) 0 0
\(85\) −3.46410 −0.375735
\(86\) −17.6603 −1.90435
\(87\) 0 0
\(88\) 2.19615 0.234111
\(89\) −0.928203 −0.0983893 −0.0491947 0.998789i \(-0.515665\pi\)
−0.0491947 + 0.998789i \(0.515665\pi\)
\(90\) 0 0
\(91\) 2.00000 0.209657
\(92\) −4.73205 −0.493350
\(93\) 0 0
\(94\) 10.3923 1.07188
\(95\) 4.19615 0.430516
\(96\) 0 0
\(97\) 2.00000 0.203069 0.101535 0.994832i \(-0.467625\pi\)
0.101535 + 0.994832i \(0.467625\pi\)
\(98\) 5.19615 0.524891
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) −12.9282 −1.28640 −0.643202 0.765696i \(-0.722395\pi\)
−0.643202 + 0.765696i \(0.722395\pi\)
\(102\) 0 0
\(103\) 10.1962 1.00466 0.502328 0.864677i \(-0.332477\pi\)
0.502328 + 0.864677i \(0.332477\pi\)
\(104\) 1.73205 0.169842
\(105\) 0 0
\(106\) −18.0000 −1.74831
\(107\) −0.339746 −0.0328445 −0.0164222 0.999865i \(-0.505228\pi\)
−0.0164222 + 0.999865i \(0.505228\pi\)
\(108\) 0 0
\(109\) 2.00000 0.191565 0.0957826 0.995402i \(-0.469465\pi\)
0.0957826 + 0.995402i \(0.469465\pi\)
\(110\) −2.19615 −0.209395
\(111\) 0 0
\(112\) −10.0000 −0.944911
\(113\) −15.4641 −1.45474 −0.727370 0.686245i \(-0.759258\pi\)
−0.727370 + 0.686245i \(0.759258\pi\)
\(114\) 0 0
\(115\) −4.73205 −0.441266
\(116\) 9.46410 0.878720
\(117\) 0 0
\(118\) −26.1962 −2.41155
\(119\) −6.92820 −0.635107
\(120\) 0 0
\(121\) −9.39230 −0.853846
\(122\) −21.4641 −1.94327
\(123\) 0 0
\(124\) −0.196152 −0.0176150
\(125\) 1.00000 0.0894427
\(126\) 0 0
\(127\) 5.80385 0.515008 0.257504 0.966277i \(-0.417100\pi\)
0.257504 + 0.966277i \(0.417100\pi\)
\(128\) −12.1244 −1.07165
\(129\) 0 0
\(130\) −1.73205 −0.151911
\(131\) 0 0 1.00000i \(-0.5\pi\)
1.00000i \(0.5\pi\)
\(132\) 0 0
\(133\) 8.39230 0.727705
\(134\) 24.9282 2.15347
\(135\) 0 0
\(136\) −6.00000 −0.514496
\(137\) 12.9282 1.10453 0.552265 0.833668i \(-0.313763\pi\)
0.552265 + 0.833668i \(0.313763\pi\)
\(138\) 0 0
\(139\) −8.39230 −0.711826 −0.355913 0.934519i \(-0.615830\pi\)
−0.355913 + 0.934519i \(0.615830\pi\)
\(140\) 2.00000 0.169031
\(141\) 0 0
\(142\) 2.19615 0.184297
\(143\) 1.26795 0.106031
\(144\) 0 0
\(145\) 9.46410 0.785951
\(146\) 6.92820 0.573382
\(147\) 0 0
\(148\) −4.00000 −0.328798
\(149\) −19.8564 −1.62670 −0.813350 0.581775i \(-0.802359\pi\)
−0.813350 + 0.581775i \(0.802359\pi\)
\(150\) 0 0
\(151\) −12.1962 −0.992509 −0.496254 0.868177i \(-0.665292\pi\)
−0.496254 + 0.868177i \(0.665292\pi\)
\(152\) 7.26795 0.589509
\(153\) 0 0
\(154\) −4.39230 −0.353942
\(155\) −0.196152 −0.0157553
\(156\) 0 0
\(157\) −10.0000 −0.798087 −0.399043 0.916932i \(-0.630658\pi\)
−0.399043 + 0.916932i \(0.630658\pi\)
\(158\) −21.4641 −1.70759
\(159\) 0 0
\(160\) 5.19615 0.410792
\(161\) −9.46410 −0.745876
\(162\) 0 0
\(163\) 6.39230 0.500684 0.250342 0.968157i \(-0.419457\pi\)
0.250342 + 0.968157i \(0.419457\pi\)
\(164\) 3.46410 0.270501
\(165\) 0 0
\(166\) −10.3923 −0.806599
\(167\) −12.9282 −1.00041 −0.500207 0.865906i \(-0.666743\pi\)
−0.500207 + 0.865906i \(0.666743\pi\)
\(168\) 0 0
\(169\) 1.00000 0.0769231
\(170\) 6.00000 0.460179
\(171\) 0 0
\(172\) 10.1962 0.777449
\(173\) −15.4641 −1.17571 −0.587857 0.808965i \(-0.700028\pi\)
−0.587857 + 0.808965i \(0.700028\pi\)
\(174\) 0 0
\(175\) 2.00000 0.151186
\(176\) −6.33975 −0.477876
\(177\) 0 0
\(178\) 1.60770 0.120502
\(179\) 5.07180 0.379084 0.189542 0.981873i \(-0.439300\pi\)
0.189542 + 0.981873i \(0.439300\pi\)
\(180\) 0 0
\(181\) −20.3923 −1.51575 −0.757874 0.652401i \(-0.773762\pi\)
−0.757874 + 0.652401i \(0.773762\pi\)
\(182\) −3.46410 −0.256776
\(183\) 0 0
\(184\) −8.19615 −0.604228
\(185\) −4.00000 −0.294086
\(186\) 0 0
\(187\) −4.39230 −0.321197
\(188\) −6.00000 −0.437595
\(189\) 0 0
\(190\) −7.26795 −0.527272
\(191\) 18.9282 1.36960 0.684798 0.728733i \(-0.259890\pi\)
0.684798 + 0.728733i \(0.259890\pi\)
\(192\) 0 0
\(193\) −10.0000 −0.719816 −0.359908 0.932988i \(-0.617192\pi\)
−0.359908 + 0.932988i \(0.617192\pi\)
\(194\) −3.46410 −0.248708
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −0.928203 −0.0661317 −0.0330659 0.999453i \(-0.510527\pi\)
−0.0330659 + 0.999453i \(0.510527\pi\)
\(198\) 0 0
\(199\) 20.0000 1.41776 0.708881 0.705328i \(-0.249200\pi\)
0.708881 + 0.705328i \(0.249200\pi\)
\(200\) 1.73205 0.122474
\(201\) 0 0
\(202\) 22.3923 1.57552
\(203\) 18.9282 1.32850
\(204\) 0 0
\(205\) 3.46410 0.241943
\(206\) −17.6603 −1.23045
\(207\) 0 0
\(208\) −5.00000 −0.346688
\(209\) 5.32051 0.368027
\(210\) 0 0
\(211\) 8.00000 0.550743 0.275371 0.961338i \(-0.411199\pi\)
0.275371 + 0.961338i \(0.411199\pi\)
\(212\) 10.3923 0.713746
\(213\) 0 0
\(214\) 0.588457 0.0402261
\(215\) 10.1962 0.695372
\(216\) 0 0
\(217\) −0.392305 −0.0266314
\(218\) −3.46410 −0.234619
\(219\) 0 0
\(220\) 1.26795 0.0854851
\(221\) −3.46410 −0.233021
\(222\) 0 0
\(223\) 2.00000 0.133930 0.0669650 0.997755i \(-0.478668\pi\)
0.0669650 + 0.997755i \(0.478668\pi\)
\(224\) 10.3923 0.694365
\(225\) 0 0
\(226\) 26.7846 1.78169
\(227\) −3.46410 −0.229920 −0.114960 0.993370i \(-0.536674\pi\)
−0.114960 + 0.993370i \(0.536674\pi\)
\(228\) 0 0
\(229\) −14.3923 −0.951070 −0.475535 0.879697i \(-0.657746\pi\)
−0.475535 + 0.879697i \(0.657746\pi\)
\(230\) 8.19615 0.540438
\(231\) 0 0
\(232\) 16.3923 1.07621
\(233\) 6.00000 0.393073 0.196537 0.980497i \(-0.437031\pi\)
0.196537 + 0.980497i \(0.437031\pi\)
\(234\) 0 0
\(235\) −6.00000 −0.391397
\(236\) 15.1244 0.984512
\(237\) 0 0
\(238\) 12.0000 0.777844
\(239\) 3.80385 0.246050 0.123025 0.992404i \(-0.460740\pi\)
0.123025 + 0.992404i \(0.460740\pi\)
\(240\) 0 0
\(241\) 18.3923 1.18475 0.592376 0.805661i \(-0.298190\pi\)
0.592376 + 0.805661i \(0.298190\pi\)
\(242\) 16.2679 1.04574
\(243\) 0 0
\(244\) 12.3923 0.793336
\(245\) −3.00000 −0.191663
\(246\) 0 0
\(247\) 4.19615 0.266995
\(248\) −0.339746 −0.0215739
\(249\) 0 0
\(250\) −1.73205 −0.109545
\(251\) 14.5359 0.917498 0.458749 0.888566i \(-0.348298\pi\)
0.458749 + 0.888566i \(0.348298\pi\)
\(252\) 0 0
\(253\) −6.00000 −0.377217
\(254\) −10.0526 −0.630754
\(255\) 0 0
\(256\) 19.0000 1.18750
\(257\) 7.85641 0.490069 0.245035 0.969514i \(-0.421201\pi\)
0.245035 + 0.969514i \(0.421201\pi\)
\(258\) 0 0
\(259\) −8.00000 −0.497096
\(260\) 1.00000 0.0620174
\(261\) 0 0
\(262\) 0 0
\(263\) −4.73205 −0.291791 −0.145895 0.989300i \(-0.546606\pi\)
−0.145895 + 0.989300i \(0.546606\pi\)
\(264\) 0 0
\(265\) 10.3923 0.638394
\(266\) −14.5359 −0.891253
\(267\) 0 0
\(268\) −14.3923 −0.879150
\(269\) −7.85641 −0.479014 −0.239507 0.970895i \(-0.576986\pi\)
−0.239507 + 0.970895i \(0.576986\pi\)
\(270\) 0 0
\(271\) −20.9808 −1.27449 −0.637245 0.770661i \(-0.719926\pi\)
−0.637245 + 0.770661i \(0.719926\pi\)
\(272\) 17.3205 1.05021
\(273\) 0 0
\(274\) −22.3923 −1.35277
\(275\) 1.26795 0.0764602
\(276\) 0 0
\(277\) −5.60770 −0.336934 −0.168467 0.985707i \(-0.553882\pi\)
−0.168467 + 0.985707i \(0.553882\pi\)
\(278\) 14.5359 0.871805
\(279\) 0 0
\(280\) 3.46410 0.207020
\(281\) −1.60770 −0.0959071 −0.0479535 0.998850i \(-0.515270\pi\)
−0.0479535 + 0.998850i \(0.515270\pi\)
\(282\) 0 0
\(283\) 1.41154 0.0839075 0.0419538 0.999120i \(-0.486642\pi\)
0.0419538 + 0.999120i \(0.486642\pi\)
\(284\) −1.26795 −0.0752389
\(285\) 0 0
\(286\) −2.19615 −0.129861
\(287\) 6.92820 0.408959
\(288\) 0 0
\(289\) −5.00000 −0.294118
\(290\) −16.3923 −0.962589
\(291\) 0 0
\(292\) −4.00000 −0.234082
\(293\) 18.9282 1.10580 0.552899 0.833248i \(-0.313522\pi\)
0.552899 + 0.833248i \(0.313522\pi\)
\(294\) 0 0
\(295\) 15.1244 0.880574
\(296\) −6.92820 −0.402694
\(297\) 0 0
\(298\) 34.3923 1.99229
\(299\) −4.73205 −0.273662
\(300\) 0 0
\(301\) 20.3923 1.17539
\(302\) 21.1244 1.21557
\(303\) 0 0
\(304\) −20.9808 −1.20333
\(305\) 12.3923 0.709581
\(306\) 0 0
\(307\) 22.7846 1.30039 0.650193 0.759769i \(-0.274688\pi\)
0.650193 + 0.759769i \(0.274688\pi\)
\(308\) 2.53590 0.144496
\(309\) 0 0
\(310\) 0.339746 0.0192963
\(311\) −4.39230 −0.249065 −0.124532 0.992216i \(-0.539743\pi\)
−0.124532 + 0.992216i \(0.539743\pi\)
\(312\) 0 0
\(313\) 6.39230 0.361314 0.180657 0.983546i \(-0.442178\pi\)
0.180657 + 0.983546i \(0.442178\pi\)
\(314\) 17.3205 0.977453
\(315\) 0 0
\(316\) 12.3923 0.697122
\(317\) −24.0000 −1.34797 −0.673987 0.738743i \(-0.735420\pi\)
−0.673987 + 0.738743i \(0.735420\pi\)
\(318\) 0 0
\(319\) 12.0000 0.671871
\(320\) 1.00000 0.0559017
\(321\) 0 0
\(322\) 16.3923 0.913507
\(323\) −14.5359 −0.808799
\(324\) 0 0
\(325\) 1.00000 0.0554700
\(326\) −11.0718 −0.613210
\(327\) 0 0
\(328\) 6.00000 0.331295
\(329\) −12.0000 −0.661581
\(330\) 0 0
\(331\) −28.5885 −1.57136 −0.785682 0.618631i \(-0.787688\pi\)
−0.785682 + 0.618631i \(0.787688\pi\)
\(332\) 6.00000 0.329293
\(333\) 0 0
\(334\) 22.3923 1.22525
\(335\) −14.3923 −0.786336
\(336\) 0 0
\(337\) −5.60770 −0.305471 −0.152735 0.988267i \(-0.548808\pi\)
−0.152735 + 0.988267i \(0.548808\pi\)
\(338\) −1.73205 −0.0942111
\(339\) 0 0
\(340\) −3.46410 −0.187867
\(341\) −0.248711 −0.0134685
\(342\) 0 0
\(343\) −20.0000 −1.07990
\(344\) 17.6603 0.952177
\(345\) 0 0
\(346\) 26.7846 1.43995
\(347\) −11.6603 −0.625955 −0.312978 0.949761i \(-0.601326\pi\)
−0.312978 + 0.949761i \(0.601326\pi\)
\(348\) 0 0
\(349\) 6.39230 0.342172 0.171086 0.985256i \(-0.445272\pi\)
0.171086 + 0.985256i \(0.445272\pi\)
\(350\) −3.46410 −0.185164
\(351\) 0 0
\(352\) 6.58846 0.351166
\(353\) −27.7128 −1.47500 −0.737502 0.675345i \(-0.763995\pi\)
−0.737502 + 0.675345i \(0.763995\pi\)
\(354\) 0 0
\(355\) −1.26795 −0.0672958
\(356\) −0.928203 −0.0491947
\(357\) 0 0
\(358\) −8.78461 −0.464281
\(359\) −8.19615 −0.432576 −0.216288 0.976330i \(-0.569395\pi\)
−0.216288 + 0.976330i \(0.569395\pi\)
\(360\) 0 0
\(361\) −1.39230 −0.0732792
\(362\) 35.3205 1.85640
\(363\) 0 0
\(364\) 2.00000 0.104828
\(365\) −4.00000 −0.209370
\(366\) 0 0
\(367\) 22.1962 1.15863 0.579315 0.815104i \(-0.303320\pi\)
0.579315 + 0.815104i \(0.303320\pi\)
\(368\) 23.6603 1.23338
\(369\) 0 0
\(370\) 6.92820 0.360180
\(371\) 20.7846 1.07908
\(372\) 0 0
\(373\) −10.0000 −0.517780 −0.258890 0.965907i \(-0.583357\pi\)
−0.258890 + 0.965907i \(0.583357\pi\)
\(374\) 7.60770 0.393385
\(375\) 0 0
\(376\) −10.3923 −0.535942
\(377\) 9.46410 0.487426
\(378\) 0 0
\(379\) −32.9808 −1.69411 −0.847054 0.531507i \(-0.821626\pi\)
−0.847054 + 0.531507i \(0.821626\pi\)
\(380\) 4.19615 0.215258
\(381\) 0 0
\(382\) −32.7846 −1.67741
\(383\) 0.928203 0.0474290 0.0237145 0.999719i \(-0.492451\pi\)
0.0237145 + 0.999719i \(0.492451\pi\)
\(384\) 0 0
\(385\) 2.53590 0.129241
\(386\) 17.3205 0.881591
\(387\) 0 0
\(388\) 2.00000 0.101535
\(389\) −6.00000 −0.304212 −0.152106 0.988364i \(-0.548606\pi\)
−0.152106 + 0.988364i \(0.548606\pi\)
\(390\) 0 0
\(391\) 16.3923 0.828994
\(392\) −5.19615 −0.262445
\(393\) 0 0
\(394\) 1.60770 0.0809945
\(395\) 12.3923 0.623525
\(396\) 0 0
\(397\) −12.7846 −0.641641 −0.320821 0.947140i \(-0.603959\pi\)
−0.320821 + 0.947140i \(0.603959\pi\)
\(398\) −34.6410 −1.73640
\(399\) 0 0
\(400\) −5.00000 −0.250000
\(401\) 23.0718 1.15215 0.576075 0.817397i \(-0.304584\pi\)
0.576075 + 0.817397i \(0.304584\pi\)
\(402\) 0 0
\(403\) −0.196152 −0.00977105
\(404\) −12.9282 −0.643202
\(405\) 0 0
\(406\) −32.7846 −1.62707
\(407\) −5.07180 −0.251400
\(408\) 0 0
\(409\) −38.3923 −1.89838 −0.949189 0.314708i \(-0.898094\pi\)
−0.949189 + 0.314708i \(0.898094\pi\)
\(410\) −6.00000 −0.296319
\(411\) 0 0
\(412\) 10.1962 0.502328
\(413\) 30.2487 1.48844
\(414\) 0 0
\(415\) 6.00000 0.294528
\(416\) 5.19615 0.254762
\(417\) 0 0
\(418\) −9.21539 −0.450739
\(419\) 9.46410 0.462352 0.231176 0.972912i \(-0.425743\pi\)
0.231176 + 0.972912i \(0.425743\pi\)
\(420\) 0 0
\(421\) 10.7846 0.525610 0.262805 0.964849i \(-0.415352\pi\)
0.262805 + 0.964849i \(0.415352\pi\)
\(422\) −13.8564 −0.674519
\(423\) 0 0
\(424\) 18.0000 0.874157
\(425\) −3.46410 −0.168034
\(426\) 0 0
\(427\) 24.7846 1.19941
\(428\) −0.339746 −0.0164222
\(429\) 0 0
\(430\) −17.6603 −0.851653
\(431\) −19.5167 −0.940084 −0.470042 0.882644i \(-0.655761\pi\)
−0.470042 + 0.882644i \(0.655761\pi\)
\(432\) 0 0
\(433\) −6.78461 −0.326048 −0.163024 0.986622i \(-0.552125\pi\)
−0.163024 + 0.986622i \(0.552125\pi\)
\(434\) 0.679492 0.0326167
\(435\) 0 0
\(436\) 2.00000 0.0957826
\(437\) −19.8564 −0.949861
\(438\) 0 0
\(439\) 32.0000 1.52728 0.763638 0.645644i \(-0.223411\pi\)
0.763638 + 0.645644i \(0.223411\pi\)
\(440\) 2.19615 0.104697
\(441\) 0 0
\(442\) 6.00000 0.285391
\(443\) −34.9808 −1.66199 −0.830993 0.556283i \(-0.812227\pi\)
−0.830993 + 0.556283i \(0.812227\pi\)
\(444\) 0 0
\(445\) −0.928203 −0.0440011
\(446\) −3.46410 −0.164030
\(447\) 0 0
\(448\) 2.00000 0.0944911
\(449\) −27.4641 −1.29611 −0.648056 0.761593i \(-0.724418\pi\)
−0.648056 + 0.761593i \(0.724418\pi\)
\(450\) 0 0
\(451\) 4.39230 0.206826
\(452\) −15.4641 −0.727370
\(453\) 0 0
\(454\) 6.00000 0.281594
\(455\) 2.00000 0.0937614
\(456\) 0 0
\(457\) −30.7846 −1.44004 −0.720022 0.693952i \(-0.755868\pi\)
−0.720022 + 0.693952i \(0.755868\pi\)
\(458\) 24.9282 1.16482
\(459\) 0 0
\(460\) −4.73205 −0.220633
\(461\) −3.46410 −0.161339 −0.0806696 0.996741i \(-0.525706\pi\)
−0.0806696 + 0.996741i \(0.525706\pi\)
\(462\) 0 0
\(463\) 18.3923 0.854763 0.427381 0.904071i \(-0.359436\pi\)
0.427381 + 0.904071i \(0.359436\pi\)
\(464\) −47.3205 −2.19680
\(465\) 0 0
\(466\) −10.3923 −0.481414
\(467\) −38.1962 −1.76751 −0.883754 0.467953i \(-0.844992\pi\)
−0.883754 + 0.467953i \(0.844992\pi\)
\(468\) 0 0
\(469\) −28.7846 −1.32915
\(470\) 10.3923 0.479361
\(471\) 0 0
\(472\) 26.1962 1.20578
\(473\) 12.9282 0.594439
\(474\) 0 0
\(475\) 4.19615 0.192533
\(476\) −6.92820 −0.317554
\(477\) 0 0
\(478\) −6.58846 −0.301349
\(479\) −18.3397 −0.837964 −0.418982 0.907994i \(-0.637613\pi\)
−0.418982 + 0.907994i \(0.637613\pi\)
\(480\) 0 0
\(481\) −4.00000 −0.182384
\(482\) −31.8564 −1.45102
\(483\) 0 0
\(484\) −9.39230 −0.426923
\(485\) 2.00000 0.0908153
\(486\) 0 0
\(487\) −5.60770 −0.254109 −0.127054 0.991896i \(-0.540552\pi\)
−0.127054 + 0.991896i \(0.540552\pi\)
\(488\) 21.4641 0.971634
\(489\) 0 0
\(490\) 5.19615 0.234738
\(491\) 9.46410 0.427109 0.213554 0.976931i \(-0.431496\pi\)
0.213554 + 0.976931i \(0.431496\pi\)
\(492\) 0 0
\(493\) −32.7846 −1.47654
\(494\) −7.26795 −0.327000
\(495\) 0 0
\(496\) 0.980762 0.0440375
\(497\) −2.53590 −0.113751
\(498\) 0 0
\(499\) 12.9808 0.581099 0.290549 0.956860i \(-0.406162\pi\)
0.290549 + 0.956860i \(0.406162\pi\)
\(500\) 1.00000 0.0447214
\(501\) 0 0
\(502\) −25.1769 −1.12370
\(503\) 25.5167 1.13773 0.568866 0.822430i \(-0.307382\pi\)
0.568866 + 0.822430i \(0.307382\pi\)
\(504\) 0 0
\(505\) −12.9282 −0.575297
\(506\) 10.3923 0.461994
\(507\) 0 0
\(508\) 5.80385 0.257504
\(509\) −32.5359 −1.44213 −0.721064 0.692868i \(-0.756347\pi\)
−0.721064 + 0.692868i \(0.756347\pi\)
\(510\) 0 0
\(511\) −8.00000 −0.353899
\(512\) −8.66025 −0.382733
\(513\) 0 0
\(514\) −13.6077 −0.600210
\(515\) 10.1962 0.449296
\(516\) 0 0
\(517\) −7.60770 −0.334586
\(518\) 13.8564 0.608816
\(519\) 0 0
\(520\) 1.73205 0.0759555
\(521\) 7.60770 0.333299 0.166650 0.986016i \(-0.446705\pi\)
0.166650 + 0.986016i \(0.446705\pi\)
\(522\) 0 0
\(523\) −13.8038 −0.603600 −0.301800 0.953371i \(-0.597588\pi\)
−0.301800 + 0.953371i \(0.597588\pi\)
\(524\) 0 0
\(525\) 0 0
\(526\) 8.19615 0.357369
\(527\) 0.679492 0.0295991
\(528\) 0 0
\(529\) −0.607695 −0.0264215
\(530\) −18.0000 −0.781870
\(531\) 0 0
\(532\) 8.39230 0.363853
\(533\) 3.46410 0.150047
\(534\) 0 0
\(535\) −0.339746 −0.0146885
\(536\) −24.9282 −1.07673
\(537\) 0 0
\(538\) 13.6077 0.586669
\(539\) −3.80385 −0.163843
\(540\) 0 0
\(541\) −5.60770 −0.241094 −0.120547 0.992708i \(-0.538465\pi\)
−0.120547 + 0.992708i \(0.538465\pi\)
\(542\) 36.3397 1.56093
\(543\) 0 0
\(544\) −18.0000 −0.771744
\(545\) 2.00000 0.0856706
\(546\) 0 0
\(547\) −1.80385 −0.0771270 −0.0385635 0.999256i \(-0.512278\pi\)
−0.0385635 + 0.999256i \(0.512278\pi\)
\(548\) 12.9282 0.552265
\(549\) 0 0
\(550\) −2.19615 −0.0936443
\(551\) 39.7128 1.69182
\(552\) 0 0
\(553\) 24.7846 1.05395
\(554\) 9.71281 0.412658
\(555\) 0 0
\(556\) −8.39230 −0.355913
\(557\) 25.8564 1.09557 0.547786 0.836619i \(-0.315471\pi\)
0.547786 + 0.836619i \(0.315471\pi\)
\(558\) 0 0
\(559\) 10.1962 0.431251
\(560\) −10.0000 −0.422577
\(561\) 0 0
\(562\) 2.78461 0.117462
\(563\) −16.0526 −0.676535 −0.338267 0.941050i \(-0.609841\pi\)
−0.338267 + 0.941050i \(0.609841\pi\)
\(564\) 0 0
\(565\) −15.4641 −0.650580
\(566\) −2.44486 −0.102765
\(567\) 0 0
\(568\) −2.19615 −0.0921485
\(569\) 9.46410 0.396756 0.198378 0.980126i \(-0.436433\pi\)
0.198378 + 0.980126i \(0.436433\pi\)
\(570\) 0 0
\(571\) 15.6077 0.653162 0.326581 0.945169i \(-0.394103\pi\)
0.326581 + 0.945169i \(0.394103\pi\)
\(572\) 1.26795 0.0530156
\(573\) 0 0
\(574\) −12.0000 −0.500870
\(575\) −4.73205 −0.197340
\(576\) 0 0
\(577\) −4.00000 −0.166522 −0.0832611 0.996528i \(-0.526534\pi\)
−0.0832611 + 0.996528i \(0.526534\pi\)
\(578\) 8.66025 0.360219
\(579\) 0 0
\(580\) 9.46410 0.392975
\(581\) 12.0000 0.497844
\(582\) 0 0
\(583\) 13.1769 0.545732
\(584\) −6.92820 −0.286691
\(585\) 0 0
\(586\) −32.7846 −1.35432
\(587\) 15.4641 0.638272 0.319136 0.947709i \(-0.396607\pi\)
0.319136 + 0.947709i \(0.396607\pi\)
\(588\) 0 0
\(589\) −0.823085 −0.0339146
\(590\) −26.1962 −1.07848
\(591\) 0 0
\(592\) 20.0000 0.821995
\(593\) 14.7846 0.607131 0.303566 0.952811i \(-0.401823\pi\)
0.303566 + 0.952811i \(0.401823\pi\)
\(594\) 0 0
\(595\) −6.92820 −0.284029
\(596\) −19.8564 −0.813350
\(597\) 0 0
\(598\) 8.19615 0.335166
\(599\) 28.3923 1.16008 0.580039 0.814589i \(-0.303037\pi\)
0.580039 + 0.814589i \(0.303037\pi\)
\(600\) 0 0
\(601\) −39.5692 −1.61406 −0.807031 0.590509i \(-0.798927\pi\)
−0.807031 + 0.590509i \(0.798927\pi\)
\(602\) −35.3205 −1.43956
\(603\) 0 0
\(604\) −12.1962 −0.496254
\(605\) −9.39230 −0.381851
\(606\) 0 0
\(607\) −26.9808 −1.09512 −0.547558 0.836768i \(-0.684442\pi\)
−0.547558 + 0.836768i \(0.684442\pi\)
\(608\) 21.8038 0.884263
\(609\) 0 0
\(610\) −21.4641 −0.869056
\(611\) −6.00000 −0.242734
\(612\) 0 0
\(613\) 26.0000 1.05013 0.525065 0.851062i \(-0.324041\pi\)
0.525065 + 0.851062i \(0.324041\pi\)
\(614\) −39.4641 −1.59264
\(615\) 0 0
\(616\) 4.39230 0.176971
\(617\) 21.7128 0.874125 0.437062 0.899431i \(-0.356019\pi\)
0.437062 + 0.899431i \(0.356019\pi\)
\(618\) 0 0
\(619\) −44.9808 −1.80793 −0.903965 0.427607i \(-0.859357\pi\)
−0.903965 + 0.427607i \(0.859357\pi\)
\(620\) −0.196152 −0.00787767
\(621\) 0 0
\(622\) 7.60770 0.305041
\(623\) −1.85641 −0.0743754
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −11.0718 −0.442518
\(627\) 0 0
\(628\) −10.0000 −0.399043
\(629\) 13.8564 0.552491
\(630\) 0 0
\(631\) 16.1962 0.644759 0.322379 0.946611i \(-0.395517\pi\)
0.322379 + 0.946611i \(0.395517\pi\)
\(632\) 21.4641 0.853796
\(633\) 0 0
\(634\) 41.5692 1.65092
\(635\) 5.80385 0.230319
\(636\) 0 0
\(637\) −3.00000 −0.118864
\(638\) −20.7846 −0.822871
\(639\) 0 0
\(640\) −12.1244 −0.479257
\(641\) 0.928203 0.0366618 0.0183309 0.999832i \(-0.494165\pi\)
0.0183309 + 0.999832i \(0.494165\pi\)
\(642\) 0 0
\(643\) 34.7846 1.37177 0.685886 0.727709i \(-0.259415\pi\)
0.685886 + 0.727709i \(0.259415\pi\)
\(644\) −9.46410 −0.372938
\(645\) 0 0
\(646\) 25.1769 0.990572
\(647\) 16.0526 0.631091 0.315546 0.948910i \(-0.397812\pi\)
0.315546 + 0.948910i \(0.397812\pi\)
\(648\) 0 0
\(649\) 19.1769 0.752760
\(650\) −1.73205 −0.0679366
\(651\) 0 0
\(652\) 6.39230 0.250342
\(653\) 19.8564 0.777041 0.388521 0.921440i \(-0.372986\pi\)
0.388521 + 0.921440i \(0.372986\pi\)
\(654\) 0 0
\(655\) 0 0
\(656\) −17.3205 −0.676252
\(657\) 0 0
\(658\) 20.7846 0.810268
\(659\) 14.5359 0.566238 0.283119 0.959085i \(-0.408631\pi\)
0.283119 + 0.959085i \(0.408631\pi\)
\(660\) 0 0
\(661\) −30.7846 −1.19738 −0.598691 0.800980i \(-0.704312\pi\)
−0.598691 + 0.800980i \(0.704312\pi\)
\(662\) 49.5167 1.92452
\(663\) 0 0
\(664\) 10.3923 0.403300
\(665\) 8.39230 0.325440
\(666\) 0 0
\(667\) −44.7846 −1.73407
\(668\) −12.9282 −0.500207
\(669\) 0 0
\(670\) 24.9282 0.963061
\(671\) 15.7128 0.606586
\(672\) 0 0
\(673\) 6.39230 0.246405 0.123203 0.992382i \(-0.460683\pi\)
0.123203 + 0.992382i \(0.460683\pi\)
\(674\) 9.71281 0.374124
\(675\) 0 0
\(676\) 1.00000 0.0384615
\(677\) 10.3923 0.399409 0.199704 0.979856i \(-0.436002\pi\)
0.199704 + 0.979856i \(0.436002\pi\)
\(678\) 0 0
\(679\) 4.00000 0.153506
\(680\) −6.00000 −0.230089
\(681\) 0 0
\(682\) 0.430781 0.0164954
\(683\) −39.4641 −1.51005 −0.755026 0.655695i \(-0.772376\pi\)
−0.755026 + 0.655695i \(0.772376\pi\)
\(684\) 0 0
\(685\) 12.9282 0.493961
\(686\) 34.6410 1.32260
\(687\) 0 0
\(688\) −50.9808 −1.94362
\(689\) 10.3923 0.395915
\(690\) 0 0
\(691\) 45.7654 1.74100 0.870498 0.492171i \(-0.163797\pi\)
0.870498 + 0.492171i \(0.163797\pi\)
\(692\) −15.4641 −0.587857
\(693\) 0 0
\(694\) 20.1962 0.766635
\(695\) −8.39230 −0.318338
\(696\) 0 0
\(697\) −12.0000 −0.454532
\(698\) −11.0718 −0.419074
\(699\) 0 0
\(700\) 2.00000 0.0755929
\(701\) −42.0000 −1.58632 −0.793159 0.609015i \(-0.791565\pi\)
−0.793159 + 0.609015i \(0.791565\pi\)
\(702\) 0 0
\(703\) −16.7846 −0.633044
\(704\) 1.26795 0.0477876
\(705\) 0 0
\(706\) 48.0000 1.80650
\(707\) −25.8564 −0.972430
\(708\) 0 0
\(709\) 9.60770 0.360825 0.180412 0.983591i \(-0.442257\pi\)
0.180412 + 0.983591i \(0.442257\pi\)
\(710\) 2.19615 0.0824201
\(711\) 0 0
\(712\) −1.60770 −0.0602509
\(713\) 0.928203 0.0347615
\(714\) 0 0
\(715\) 1.26795 0.0474186
\(716\) 5.07180 0.189542
\(717\) 0 0
\(718\) 14.1962 0.529796
\(719\) −1.85641 −0.0692323 −0.0346161 0.999401i \(-0.511021\pi\)
−0.0346161 + 0.999401i \(0.511021\pi\)
\(720\) 0 0
\(721\) 20.3923 0.759449
\(722\) 2.41154 0.0897483
\(723\) 0 0
\(724\) −20.3923 −0.757874
\(725\) 9.46410 0.351488
\(726\) 0 0
\(727\) 13.4115 0.497407 0.248703 0.968580i \(-0.419996\pi\)
0.248703 + 0.968580i \(0.419996\pi\)
\(728\) 3.46410 0.128388
\(729\) 0 0
\(730\) 6.92820 0.256424
\(731\) −35.3205 −1.30638
\(732\) 0 0
\(733\) 38.0000 1.40356 0.701781 0.712393i \(-0.252388\pi\)
0.701781 + 0.712393i \(0.252388\pi\)
\(734\) −38.4449 −1.41903
\(735\) 0 0
\(736\) −24.5885 −0.906343
\(737\) −18.2487 −0.672200
\(738\) 0 0
\(739\) −7.80385 −0.287069 −0.143535 0.989645i \(-0.545847\pi\)
−0.143535 + 0.989645i \(0.545847\pi\)
\(740\) −4.00000 −0.147043
\(741\) 0 0
\(742\) −36.0000 −1.32160
\(743\) 43.8564 1.60894 0.804468 0.593996i \(-0.202451\pi\)
0.804468 + 0.593996i \(0.202451\pi\)
\(744\) 0 0
\(745\) −19.8564 −0.727482
\(746\) 17.3205 0.634149
\(747\) 0 0
\(748\) −4.39230 −0.160599
\(749\) −0.679492 −0.0248281
\(750\) 0 0
\(751\) 15.6077 0.569533 0.284766 0.958597i \(-0.408084\pi\)
0.284766 + 0.958597i \(0.408084\pi\)
\(752\) 30.0000 1.09399
\(753\) 0 0
\(754\) −16.3923 −0.596973
\(755\) −12.1962 −0.443863
\(756\) 0 0
\(757\) 18.3923 0.668480 0.334240 0.942488i \(-0.391520\pi\)
0.334240 + 0.942488i \(0.391520\pi\)
\(758\) 57.1244 2.07485
\(759\) 0 0
\(760\) 7.26795 0.263636
\(761\) −7.85641 −0.284795 −0.142397 0.989810i \(-0.545481\pi\)
−0.142397 + 0.989810i \(0.545481\pi\)
\(762\) 0 0
\(763\) 4.00000 0.144810
\(764\) 18.9282 0.684798
\(765\) 0 0
\(766\) −1.60770 −0.0580884
\(767\) 15.1244 0.546109
\(768\) 0 0
\(769\) −6.78461 −0.244659 −0.122330 0.992490i \(-0.539037\pi\)
−0.122330 + 0.992490i \(0.539037\pi\)
\(770\) −4.39230 −0.158288
\(771\) 0 0
\(772\) −10.0000 −0.359908
\(773\) 6.92820 0.249190 0.124595 0.992208i \(-0.460237\pi\)
0.124595 + 0.992208i \(0.460237\pi\)
\(774\) 0 0
\(775\) −0.196152 −0.00704600
\(776\) 3.46410 0.124354
\(777\) 0 0
\(778\) 10.3923 0.372582
\(779\) 14.5359 0.520803
\(780\) 0 0
\(781\) −1.60770 −0.0575279
\(782\) −28.3923 −1.01531
\(783\) 0 0
\(784\) 15.0000 0.535714
\(785\) −10.0000 −0.356915
\(786\) 0 0
\(787\) −51.5692 −1.83824 −0.919122 0.393973i \(-0.871100\pi\)
−0.919122 + 0.393973i \(0.871100\pi\)
\(788\) −0.928203 −0.0330659
\(789\) 0 0
\(790\) −21.4641 −0.763658
\(791\) −30.9282 −1.09968
\(792\) 0 0
\(793\) 12.3923 0.440064
\(794\) 22.1436 0.785847
\(795\) 0 0
\(796\) 20.0000 0.708881
\(797\) −28.6410 −1.01452 −0.507258 0.861794i \(-0.669341\pi\)
−0.507258 + 0.861794i \(0.669341\pi\)
\(798\) 0 0
\(799\) 20.7846 0.735307
\(800\) 5.19615 0.183712
\(801\) 0 0
\(802\) −39.9615 −1.41109
\(803\) −5.07180 −0.178980
\(804\) 0 0
\(805\) −9.46410 −0.333566
\(806\) 0.339746 0.0119670
\(807\) 0 0
\(808\) −22.3923 −0.787759
\(809\) 9.46410 0.332740 0.166370 0.986063i \(-0.446795\pi\)
0.166370 + 0.986063i \(0.446795\pi\)
\(810\) 0 0
\(811\) 28.1962 0.990101 0.495050 0.868864i \(-0.335150\pi\)
0.495050 + 0.868864i \(0.335150\pi\)
\(812\) 18.9282 0.664250
\(813\) 0 0
\(814\) 8.78461 0.307900
\(815\) 6.39230 0.223913
\(816\) 0 0
\(817\) 42.7846 1.49684
\(818\) 66.4974 2.32503
\(819\) 0 0
\(820\) 3.46410 0.120972
\(821\) 40.6410 1.41838 0.709191 0.705017i \(-0.249061\pi\)
0.709191 + 0.705017i \(0.249061\pi\)
\(822\) 0 0
\(823\) −46.5885 −1.62397 −0.811986 0.583677i \(-0.801613\pi\)
−0.811986 + 0.583677i \(0.801613\pi\)
\(824\) 17.6603 0.615224
\(825\) 0 0
\(826\) −52.3923 −1.82296
\(827\) −18.0000 −0.625921 −0.312961 0.949766i \(-0.601321\pi\)
−0.312961 + 0.949766i \(0.601321\pi\)
\(828\) 0 0
\(829\) −20.3923 −0.708254 −0.354127 0.935197i \(-0.615222\pi\)
−0.354127 + 0.935197i \(0.615222\pi\)
\(830\) −10.3923 −0.360722
\(831\) 0 0
\(832\) 1.00000 0.0346688
\(833\) 10.3923 0.360072
\(834\) 0 0
\(835\) −12.9282 −0.447399
\(836\) 5.32051 0.184014
\(837\) 0 0
\(838\) −16.3923 −0.566263
\(839\) −17.6603 −0.609700 −0.304850 0.952400i \(-0.598606\pi\)
−0.304850 + 0.952400i \(0.598606\pi\)
\(840\) 0 0
\(841\) 60.5692 2.08859
\(842\) −18.6795 −0.643738
\(843\) 0 0
\(844\) 8.00000 0.275371
\(845\) 1.00000 0.0344010
\(846\) 0 0
\(847\) −18.7846 −0.645447
\(848\) −51.9615 −1.78437
\(849\) 0 0
\(850\) 6.00000 0.205798
\(851\) 18.9282 0.648850
\(852\) 0 0
\(853\) 8.00000 0.273915 0.136957 0.990577i \(-0.456268\pi\)
0.136957 + 0.990577i \(0.456268\pi\)
\(854\) −42.9282 −1.46897
\(855\) 0 0
\(856\) −0.588457 −0.0201131
\(857\) −47.5692 −1.62493 −0.812467 0.583007i \(-0.801876\pi\)
−0.812467 + 0.583007i \(0.801876\pi\)
\(858\) 0 0
\(859\) 45.1769 1.54142 0.770708 0.637188i \(-0.219903\pi\)
0.770708 + 0.637188i \(0.219903\pi\)
\(860\) 10.1962 0.347686
\(861\) 0 0
\(862\) 33.8038 1.15136
\(863\) −2.78461 −0.0947892 −0.0473946 0.998876i \(-0.515092\pi\)
−0.0473946 + 0.998876i \(0.515092\pi\)
\(864\) 0 0
\(865\) −15.4641 −0.525795
\(866\) 11.7513 0.399325
\(867\) 0 0
\(868\) −0.392305 −0.0133157
\(869\) 15.7128 0.533021
\(870\) 0 0
\(871\) −14.3923 −0.487665
\(872\) 3.46410 0.117309
\(873\) 0 0
\(874\) 34.3923 1.16334
\(875\) 2.00000 0.0676123
\(876\) 0 0
\(877\) 2.00000 0.0675352 0.0337676 0.999430i \(-0.489249\pi\)
0.0337676 + 0.999430i \(0.489249\pi\)
\(878\) −55.4256 −1.87052
\(879\) 0 0
\(880\) −6.33975 −0.213713
\(881\) 12.6795 0.427183 0.213591 0.976923i \(-0.431484\pi\)
0.213591 + 0.976923i \(0.431484\pi\)
\(882\) 0 0
\(883\) 34.1962 1.15079 0.575396 0.817875i \(-0.304848\pi\)
0.575396 + 0.817875i \(0.304848\pi\)
\(884\) −3.46410 −0.116510
\(885\) 0 0
\(886\) 60.5885 2.03551
\(887\) −17.9090 −0.601324 −0.300662 0.953731i \(-0.597208\pi\)
−0.300662 + 0.953731i \(0.597208\pi\)
\(888\) 0 0
\(889\) 11.6077 0.389310
\(890\) 1.60770 0.0538901
\(891\) 0 0
\(892\) 2.00000 0.0669650
\(893\) −25.1769 −0.842513
\(894\) 0 0
\(895\) 5.07180 0.169531
\(896\) −24.2487 −0.810093
\(897\) 0 0
\(898\) 47.5692 1.58741
\(899\) −1.85641 −0.0619146
\(900\) 0 0
\(901\) −36.0000 −1.19933
\(902\) −7.60770 −0.253309
\(903\) 0 0
\(904\) −26.7846 −0.890843
\(905\) −20.3923 −0.677863
\(906\) 0 0
\(907\) 39.7654 1.32039 0.660194 0.751095i \(-0.270474\pi\)
0.660194 + 0.751095i \(0.270474\pi\)
\(908\) −3.46410 −0.114960
\(909\) 0 0
\(910\) −3.46410 −0.114834
\(911\) 36.0000 1.19273 0.596367 0.802712i \(-0.296610\pi\)
0.596367 + 0.802712i \(0.296610\pi\)
\(912\) 0 0
\(913\) 7.60770 0.251778
\(914\) 53.3205 1.76369
\(915\) 0 0
\(916\) −14.3923 −0.475535
\(917\) 0 0
\(918\) 0 0
\(919\) −53.1769 −1.75414 −0.877072 0.480358i \(-0.840507\pi\)
−0.877072 + 0.480358i \(0.840507\pi\)
\(920\) −8.19615 −0.270219
\(921\) 0 0
\(922\) 6.00000 0.197599
\(923\) −1.26795 −0.0417351
\(924\) 0 0
\(925\) −4.00000 −0.131519
\(926\) −31.8564 −1.04687
\(927\) 0 0
\(928\) 49.1769 1.61431
\(929\) −51.4641 −1.68848 −0.844241 0.535963i \(-0.819948\pi\)
−0.844241 + 0.535963i \(0.819948\pi\)
\(930\) 0 0
\(931\) −12.5885 −0.412570
\(932\) 6.00000 0.196537
\(933\) 0 0
\(934\) 66.1577 2.16475
\(935\) −4.39230 −0.143644
\(936\) 0 0
\(937\) −6.78461 −0.221644 −0.110822 0.993840i \(-0.535348\pi\)
−0.110822 + 0.993840i \(0.535348\pi\)
\(938\) 49.8564 1.62787
\(939\) 0 0
\(940\) −6.00000 −0.195698
\(941\) 31.1769 1.01634 0.508169 0.861257i \(-0.330322\pi\)
0.508169 + 0.861257i \(0.330322\pi\)
\(942\) 0 0
\(943\) −16.3923 −0.533807
\(944\) −75.6218 −2.46128
\(945\) 0 0
\(946\) −22.3923 −0.728037
\(947\) 28.6410 0.930708 0.465354 0.885125i \(-0.345927\pi\)
0.465354 + 0.885125i \(0.345927\pi\)
\(948\) 0 0
\(949\) −4.00000 −0.129845
\(950\) −7.26795 −0.235803
\(951\) 0 0
\(952\) −12.0000 −0.388922
\(953\) 12.9282 0.418786 0.209393 0.977832i \(-0.432851\pi\)
0.209393 + 0.977832i \(0.432851\pi\)
\(954\) 0 0
\(955\) 18.9282 0.612502
\(956\) 3.80385 0.123025
\(957\) 0 0
\(958\) 31.7654 1.02629
\(959\) 25.8564 0.834947
\(960\) 0 0
\(961\) −30.9615 −0.998759
\(962\) 6.92820 0.223374
\(963\) 0 0
\(964\) 18.3923 0.592376
\(965\) −10.0000 −0.321911
\(966\) 0 0
\(967\) −29.6077 −0.952119 −0.476060 0.879413i \(-0.657935\pi\)
−0.476060 + 0.879413i \(0.657935\pi\)
\(968\) −16.2679 −0.522872
\(969\) 0 0
\(970\) −3.46410 −0.111226
\(971\) −5.07180 −0.162762 −0.0813809 0.996683i \(-0.525933\pi\)
−0.0813809 + 0.996683i \(0.525933\pi\)
\(972\) 0 0
\(973\) −16.7846 −0.538090
\(974\) 9.71281 0.311219
\(975\) 0 0
\(976\) −61.9615 −1.98334
\(977\) −39.7128 −1.27053 −0.635263 0.772296i \(-0.719108\pi\)
−0.635263 + 0.772296i \(0.719108\pi\)
\(978\) 0 0
\(979\) −1.17691 −0.0376144
\(980\) −3.00000 −0.0958315
\(981\) 0 0
\(982\) −16.3923 −0.523099
\(983\) 13.6077 0.434018 0.217009 0.976170i \(-0.430370\pi\)
0.217009 + 0.976170i \(0.430370\pi\)
\(984\) 0 0
\(985\) −0.928203 −0.0295750
\(986\) 56.7846 1.80839
\(987\) 0 0
\(988\) 4.19615 0.133497
\(989\) −48.2487 −1.53422
\(990\) 0 0
\(991\) 8.00000 0.254128 0.127064 0.991894i \(-0.459445\pi\)
0.127064 + 0.991894i \(0.459445\pi\)
\(992\) −1.01924 −0.0323608
\(993\) 0 0
\(994\) 4.39230 0.139315
\(995\) 20.0000 0.634043
\(996\) 0 0
\(997\) 54.3923 1.72262 0.861311 0.508078i \(-0.169644\pi\)
0.861311 + 0.508078i \(0.169644\pi\)
\(998\) −22.4833 −0.711698
\(999\) 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 585.2.a.k.1.1 2
3.2 odd 2 65.2.a.c.1.2 2
4.3 odd 2 9360.2.a.cm.1.2 2
5.2 odd 4 2925.2.c.v.2224.2 4
5.3 odd 4 2925.2.c.v.2224.3 4
5.4 even 2 2925.2.a.z.1.2 2
12.11 even 2 1040.2.a.h.1.2 2
13.12 even 2 7605.2.a.be.1.2 2
15.2 even 4 325.2.b.e.274.3 4
15.8 even 4 325.2.b.e.274.2 4
15.14 odd 2 325.2.a.g.1.1 2
21.20 even 2 3185.2.a.k.1.2 2
24.5 odd 2 4160.2.a.y.1.2 2
24.11 even 2 4160.2.a.bj.1.1 2
33.32 even 2 7865.2.a.h.1.1 2
39.2 even 12 845.2.m.c.316.2 4
39.5 even 4 845.2.c.e.506.1 4
39.8 even 4 845.2.c.e.506.3 4
39.11 even 12 845.2.m.a.316.2 4
39.17 odd 6 845.2.e.f.146.2 4
39.20 even 12 845.2.m.c.361.2 4
39.23 odd 6 845.2.e.f.191.2 4
39.29 odd 6 845.2.e.e.191.1 4
39.32 even 12 845.2.m.a.361.2 4
39.35 odd 6 845.2.e.e.146.1 4
39.38 odd 2 845.2.a.d.1.1 2
60.59 even 2 5200.2.a.ca.1.1 2
195.194 odd 2 4225.2.a.w.1.2 2
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
65.2.a.c.1.2 2 3.2 odd 2
325.2.a.g.1.1 2 15.14 odd 2
325.2.b.e.274.2 4 15.8 even 4
325.2.b.e.274.3 4 15.2 even 4
585.2.a.k.1.1 2 1.1 even 1 trivial
845.2.a.d.1.1 2 39.38 odd 2
845.2.c.e.506.1 4 39.5 even 4
845.2.c.e.506.3 4 39.8 even 4
845.2.e.e.146.1 4 39.35 odd 6
845.2.e.e.191.1 4 39.29 odd 6
845.2.e.f.146.2 4 39.17 odd 6
845.2.e.f.191.2 4 39.23 odd 6
845.2.m.a.316.2 4 39.11 even 12
845.2.m.a.361.2 4 39.32 even 12
845.2.m.c.316.2 4 39.2 even 12
845.2.m.c.361.2 4 39.20 even 12
1040.2.a.h.1.2 2 12.11 even 2
2925.2.a.z.1.2 2 5.4 even 2
2925.2.c.v.2224.2 4 5.2 odd 4
2925.2.c.v.2224.3 4 5.3 odd 4
3185.2.a.k.1.2 2 21.20 even 2
4160.2.a.y.1.2 2 24.5 odd 2
4160.2.a.bj.1.1 2 24.11 even 2
4225.2.a.w.1.2 2 195.194 odd 2
5200.2.a.ca.1.1 2 60.59 even 2
7605.2.a.be.1.2 2 13.12 even 2
7865.2.a.h.1.1 2 33.32 even 2
9360.2.a.cm.1.2 2 4.3 odd 2