Properties

Label 3450.2.a.br.1.2
Level $3450$
Weight $2$
Character 3450.1
Self dual yes
Analytic conductor $27.548$
Analytic rank $1$
Dimension $3$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 3450 = 2 \cdot 3 \cdot 5^{2} \cdot 23 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3450.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(27.5483886973\)
Analytic rank: \(1\)
Dimension: \(3\)
Coefficient field: 3.3.568.1
Defining polynomial: \(x^{3} - x^{2} - 6 x - 2\)
Coefficient ring: \(\Z[a_1, \ldots, a_{11}]\)
Coefficient ring index: \( 2 \)
Twist minimal: no (minimal twist has level 690)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(-1.76156\) of defining polynomial
Character \(\chi\) \(=\) 3450.1

$q$-expansion

\(f(q)\) \(=\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q+1.00000 q^{2} -1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{6} -2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -0.864641 q^{11} -1.00000 q^{12} -5.52311 q^{13} -2.00000 q^{14} +1.00000 q^{16} +3.52311 q^{17} +1.00000 q^{18} +8.11704 q^{19} +2.00000 q^{21} -0.864641 q^{22} -1.00000 q^{23} -1.00000 q^{24} -5.52311 q^{26} -1.00000 q^{27} -2.00000 q^{28} -2.00000 q^{29} -3.25240 q^{31} +1.00000 q^{32} +0.864641 q^{33} +3.52311 q^{34} +1.00000 q^{36} -5.13536 q^{37} +8.11704 q^{38} +5.52311 q^{39} +3.52311 q^{41} +2.00000 q^{42} -1.13536 q^{43} -0.864641 q^{44} -1.00000 q^{46} -5.52311 q^{47} -1.00000 q^{48} -3.00000 q^{49} -3.52311 q^{51} -5.52311 q^{52} -1.34153 q^{53} -1.00000 q^{54} -2.00000 q^{56} -8.11704 q^{57} -2.00000 q^{58} -10.9817 q^{59} -5.91087 q^{61} -3.25240 q^{62} -2.00000 q^{63} +1.00000 q^{64} +0.864641 q^{66} +5.91087 q^{67} +3.52311 q^{68} +1.00000 q^{69} -5.79383 q^{71} +1.00000 q^{72} -15.2524 q^{73} -5.13536 q^{74} +8.11704 q^{76} +1.72928 q^{77} +5.52311 q^{78} -3.79383 q^{79} +1.00000 q^{81} +3.52311 q^{82} -8.32320 q^{83} +2.00000 q^{84} -1.13536 q^{86} +2.00000 q^{87} -0.864641 q^{88} +4.77551 q^{89} +11.0462 q^{91} -1.00000 q^{92} +3.25240 q^{93} -5.52311 q^{94} -1.00000 q^{96} +1.04623 q^{97} -3.00000 q^{98} -0.864641 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - 6q^{7} + 3q^{8} + 3q^{9} + O(q^{10}) \) \( 3q + 3q^{2} - 3q^{3} + 3q^{4} - 3q^{6} - 6q^{7} + 3q^{8} + 3q^{9} - 3q^{12} - 4q^{13} - 6q^{14} + 3q^{16} - 2q^{17} + 3q^{18} + 4q^{19} + 6q^{21} - 3q^{23} - 3q^{24} - 4q^{26} - 3q^{27} - 6q^{28} - 6q^{29} + 8q^{31} + 3q^{32} - 2q^{34} + 3q^{36} - 18q^{37} + 4q^{38} + 4q^{39} - 2q^{41} + 6q^{42} - 6q^{43} - 3q^{46} - 4q^{47} - 3q^{48} - 9q^{49} + 2q^{51} - 4q^{52} - 14q^{53} - 3q^{54} - 6q^{56} - 4q^{57} - 6q^{58} - 10q^{59} + 10q^{61} + 8q^{62} - 6q^{63} + 3q^{64} - 10q^{67} - 2q^{68} + 3q^{69} - 10q^{71} + 3q^{72} - 28q^{73} - 18q^{74} + 4q^{76} + 4q^{78} - 4q^{79} + 3q^{81} - 2q^{82} - 12q^{83} + 6q^{84} - 6q^{86} + 6q^{87} - 16q^{89} + 8q^{91} - 3q^{92} - 8q^{93} - 4q^{94} - 3q^{96} - 22q^{97} - 9q^{98} + O(q^{100}) \)

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
\(3\) −1.00000 −0.577350
\(4\) 1.00000 0.500000
\(5\) 0 0
\(6\) −1.00000 −0.408248
\(7\) −2.00000 −0.755929 −0.377964 0.925820i \(-0.623376\pi\)
−0.377964 + 0.925820i \(0.623376\pi\)
\(8\) 1.00000 0.353553
\(9\) 1.00000 0.333333
\(10\) 0 0
\(11\) −0.864641 −0.260699 −0.130350 0.991468i \(-0.541610\pi\)
−0.130350 + 0.991468i \(0.541610\pi\)
\(12\) −1.00000 −0.288675
\(13\) −5.52311 −1.53184 −0.765918 0.642938i \(-0.777715\pi\)
−0.765918 + 0.642938i \(0.777715\pi\)
\(14\) −2.00000 −0.534522
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) 3.52311 0.854481 0.427240 0.904138i \(-0.359486\pi\)
0.427240 + 0.904138i \(0.359486\pi\)
\(18\) 1.00000 0.235702
\(19\) 8.11704 1.86218 0.931088 0.364795i \(-0.118861\pi\)
0.931088 + 0.364795i \(0.118861\pi\)
\(20\) 0 0
\(21\) 2.00000 0.436436
\(22\) −0.864641 −0.184342
\(23\) −1.00000 −0.208514
\(24\) −1.00000 −0.204124
\(25\) 0 0
\(26\) −5.52311 −1.08317
\(27\) −1.00000 −0.192450
\(28\) −2.00000 −0.377964
\(29\) −2.00000 −0.371391 −0.185695 0.982607i \(-0.559454\pi\)
−0.185695 + 0.982607i \(0.559454\pi\)
\(30\) 0 0
\(31\) −3.25240 −0.584148 −0.292074 0.956396i \(-0.594345\pi\)
−0.292074 + 0.956396i \(0.594345\pi\)
\(32\) 1.00000 0.176777
\(33\) 0.864641 0.150515
\(34\) 3.52311 0.604209
\(35\) 0 0
\(36\) 1.00000 0.166667
\(37\) −5.13536 −0.844248 −0.422124 0.906538i \(-0.638715\pi\)
−0.422124 + 0.906538i \(0.638715\pi\)
\(38\) 8.11704 1.31676
\(39\) 5.52311 0.884406
\(40\) 0 0
\(41\) 3.52311 0.550218 0.275109 0.961413i \(-0.411286\pi\)
0.275109 + 0.961413i \(0.411286\pi\)
\(42\) 2.00000 0.308607
\(43\) −1.13536 −0.173141 −0.0865703 0.996246i \(-0.527591\pi\)
−0.0865703 + 0.996246i \(0.527591\pi\)
\(44\) −0.864641 −0.130350
\(45\) 0 0
\(46\) −1.00000 −0.147442
\(47\) −5.52311 −0.805629 −0.402815 0.915282i \(-0.631968\pi\)
−0.402815 + 0.915282i \(0.631968\pi\)
\(48\) −1.00000 −0.144338
\(49\) −3.00000 −0.428571
\(50\) 0 0
\(51\) −3.52311 −0.493335
\(52\) −5.52311 −0.765918
\(53\) −1.34153 −0.184273 −0.0921364 0.995746i \(-0.529370\pi\)
−0.0921364 + 0.995746i \(0.529370\pi\)
\(54\) −1.00000 −0.136083
\(55\) 0 0
\(56\) −2.00000 −0.267261
\(57\) −8.11704 −1.07513
\(58\) −2.00000 −0.262613
\(59\) −10.9817 −1.42969 −0.714846 0.699282i \(-0.753503\pi\)
−0.714846 + 0.699282i \(0.753503\pi\)
\(60\) 0 0
\(61\) −5.91087 −0.756809 −0.378405 0.925640i \(-0.623527\pi\)
−0.378405 + 0.925640i \(0.623527\pi\)
\(62\) −3.25240 −0.413055
\(63\) −2.00000 −0.251976
\(64\) 1.00000 0.125000
\(65\) 0 0
\(66\) 0.864641 0.106430
\(67\) 5.91087 0.722128 0.361064 0.932541i \(-0.382414\pi\)
0.361064 + 0.932541i \(0.382414\pi\)
\(68\) 3.52311 0.427240
\(69\) 1.00000 0.120386
\(70\) 0 0
\(71\) −5.79383 −0.687601 −0.343801 0.939043i \(-0.611714\pi\)
−0.343801 + 0.939043i \(0.611714\pi\)
\(72\) 1.00000 0.117851
\(73\) −15.2524 −1.78516 −0.892579 0.450891i \(-0.851106\pi\)
−0.892579 + 0.450891i \(0.851106\pi\)
\(74\) −5.13536 −0.596973
\(75\) 0 0
\(76\) 8.11704 0.931088
\(77\) 1.72928 0.197070
\(78\) 5.52311 0.625370
\(79\) −3.79383 −0.426840 −0.213420 0.976961i \(-0.568460\pi\)
−0.213420 + 0.976961i \(0.568460\pi\)
\(80\) 0 0
\(81\) 1.00000 0.111111
\(82\) 3.52311 0.389063
\(83\) −8.32320 −0.913590 −0.456795 0.889572i \(-0.651003\pi\)
−0.456795 + 0.889572i \(0.651003\pi\)
\(84\) 2.00000 0.218218
\(85\) 0 0
\(86\) −1.13536 −0.122429
\(87\) 2.00000 0.214423
\(88\) −0.864641 −0.0921710
\(89\) 4.77551 0.506203 0.253102 0.967440i \(-0.418549\pi\)
0.253102 + 0.967440i \(0.418549\pi\)
\(90\) 0 0
\(91\) 11.0462 1.15796
\(92\) −1.00000 −0.104257
\(93\) 3.25240 0.337258
\(94\) −5.52311 −0.569666
\(95\) 0 0
\(96\) −1.00000 −0.102062
\(97\) 1.04623 0.106228 0.0531142 0.998588i \(-0.483085\pi\)
0.0531142 + 0.998588i \(0.483085\pi\)
\(98\) −3.00000 −0.303046
\(99\) −0.864641 −0.0868997
\(100\) 0 0
\(101\) 9.04623 0.900133 0.450067 0.892995i \(-0.351400\pi\)
0.450067 + 0.892995i \(0.351400\pi\)
\(102\) −3.52311 −0.348840
\(103\) −11.3169 −1.11509 −0.557546 0.830146i \(-0.688257\pi\)
−0.557546 + 0.830146i \(0.688257\pi\)
\(104\) −5.52311 −0.541586
\(105\) 0 0
\(106\) −1.34153 −0.130301
\(107\) −5.64015 −0.545254 −0.272627 0.962120i \(-0.587892\pi\)
−0.272627 + 0.962120i \(0.587892\pi\)
\(108\) −1.00000 −0.0962250
\(109\) −12.1816 −1.16678 −0.583392 0.812191i \(-0.698275\pi\)
−0.583392 + 0.812191i \(0.698275\pi\)
\(110\) 0 0
\(111\) 5.13536 0.487427
\(112\) −2.00000 −0.188982
\(113\) −5.79383 −0.545038 −0.272519 0.962150i \(-0.587857\pi\)
−0.272519 + 0.962150i \(0.587857\pi\)
\(114\) −8.11704 −0.760230
\(115\) 0 0
\(116\) −2.00000 −0.185695
\(117\) −5.52311 −0.510612
\(118\) −10.9817 −1.01095
\(119\) −7.04623 −0.645927
\(120\) 0 0
\(121\) −10.2524 −0.932036
\(122\) −5.91087 −0.535145
\(123\) −3.52311 −0.317669
\(124\) −3.25240 −0.292074
\(125\) 0 0
\(126\) −2.00000 −0.178174
\(127\) −0.747604 −0.0663391 −0.0331696 0.999450i \(-0.510560\pi\)
−0.0331696 + 0.999450i \(0.510560\pi\)
\(128\) 1.00000 0.0883883
\(129\) 1.13536 0.0999628
\(130\) 0 0
\(131\) 19.7572 1.72619 0.863097 0.505039i \(-0.168522\pi\)
0.863097 + 0.505039i \(0.168522\pi\)
\(132\) 0.864641 0.0752573
\(133\) −16.2341 −1.40767
\(134\) 5.91087 0.510621
\(135\) 0 0
\(136\) 3.52311 0.302105
\(137\) −0.476886 −0.0407431 −0.0203715 0.999792i \(-0.506485\pi\)
−0.0203715 + 0.999792i \(0.506485\pi\)
\(138\) 1.00000 0.0851257
\(139\) −3.45856 −0.293352 −0.146676 0.989185i \(-0.546857\pi\)
−0.146676 + 0.989185i \(0.546857\pi\)
\(140\) 0 0
\(141\) 5.52311 0.465130
\(142\) −5.79383 −0.486208
\(143\) 4.77551 0.399348
\(144\) 1.00000 0.0833333
\(145\) 0 0
\(146\) −15.2524 −1.26230
\(147\) 3.00000 0.247436
\(148\) −5.13536 −0.422124
\(149\) −8.86464 −0.726220 −0.363110 0.931746i \(-0.618285\pi\)
−0.363110 + 0.931746i \(0.618285\pi\)
\(150\) 0 0
\(151\) −6.71096 −0.546130 −0.273065 0.961996i \(-0.588037\pi\)
−0.273065 + 0.961996i \(0.588037\pi\)
\(152\) 8.11704 0.658379
\(153\) 3.52311 0.284827
\(154\) 1.72928 0.139349
\(155\) 0 0
\(156\) 5.52311 0.442203
\(157\) 14.6864 1.17210 0.586050 0.810275i \(-0.300682\pi\)
0.586050 + 0.810275i \(0.300682\pi\)
\(158\) −3.79383 −0.301821
\(159\) 1.34153 0.106390
\(160\) 0 0
\(161\) 2.00000 0.157622
\(162\) 1.00000 0.0785674
\(163\) 23.2803 1.82345 0.911727 0.410797i \(-0.134749\pi\)
0.911727 + 0.410797i \(0.134749\pi\)
\(164\) 3.52311 0.275109
\(165\) 0 0
\(166\) −8.32320 −0.646006
\(167\) 5.93545 0.459299 0.229649 0.973273i \(-0.426242\pi\)
0.229649 + 0.973273i \(0.426242\pi\)
\(168\) 2.00000 0.154303
\(169\) 17.5048 1.34652
\(170\) 0 0
\(171\) 8.11704 0.620725
\(172\) −1.13536 −0.0865703
\(173\) −25.8217 −1.96319 −0.981595 0.190973i \(-0.938836\pi\)
−0.981595 + 0.190973i \(0.938836\pi\)
\(174\) 2.00000 0.151620
\(175\) 0 0
\(176\) −0.864641 −0.0651748
\(177\) 10.9817 0.825433
\(178\) 4.77551 0.357940
\(179\) −23.3449 −1.74488 −0.872438 0.488725i \(-0.837462\pi\)
−0.872438 + 0.488725i \(0.837462\pi\)
\(180\) 0 0
\(181\) 13.3694 0.993742 0.496871 0.867824i \(-0.334482\pi\)
0.496871 + 0.867824i \(0.334482\pi\)
\(182\) 11.0462 0.818801
\(183\) 5.91087 0.436944
\(184\) −1.00000 −0.0737210
\(185\) 0 0
\(186\) 3.25240 0.238477
\(187\) −3.04623 −0.222762
\(188\) −5.52311 −0.402815
\(189\) 2.00000 0.145479
\(190\) 0 0
\(191\) 12.2341 0.885227 0.442613 0.896713i \(-0.354051\pi\)
0.442613 + 0.896713i \(0.354051\pi\)
\(192\) −1.00000 −0.0721688
\(193\) −7.45856 −0.536879 −0.268440 0.963297i \(-0.586508\pi\)
−0.268440 + 0.963297i \(0.586508\pi\)
\(194\) 1.04623 0.0751148
\(195\) 0 0
\(196\) −3.00000 −0.214286
\(197\) −2.00000 −0.142494 −0.0712470 0.997459i \(-0.522698\pi\)
−0.0712470 + 0.997459i \(0.522698\pi\)
\(198\) −0.864641 −0.0614474
\(199\) 22.8401 1.61909 0.809545 0.587059i \(-0.199714\pi\)
0.809545 + 0.587059i \(0.199714\pi\)
\(200\) 0 0
\(201\) −5.91087 −0.416921
\(202\) 9.04623 0.636490
\(203\) 4.00000 0.280745
\(204\) −3.52311 −0.246667
\(205\) 0 0
\(206\) −11.3169 −0.788489
\(207\) −1.00000 −0.0695048
\(208\) −5.52311 −0.382959
\(209\) −7.01832 −0.485467
\(210\) 0 0
\(211\) 24.2341 1.66834 0.834171 0.551506i \(-0.185946\pi\)
0.834171 + 0.551506i \(0.185946\pi\)
\(212\) −1.34153 −0.0921364
\(213\) 5.79383 0.396987
\(214\) −5.64015 −0.385553
\(215\) 0 0
\(216\) −1.00000 −0.0680414
\(217\) 6.50479 0.441574
\(218\) −12.1816 −0.825041
\(219\) 15.2524 1.03066
\(220\) 0 0
\(221\) −19.4586 −1.30892
\(222\) 5.13536 0.344663
\(223\) −13.9634 −0.935055 −0.467528 0.883978i \(-0.654855\pi\)
−0.467528 + 0.883978i \(0.654855\pi\)
\(224\) −2.00000 −0.133631
\(225\) 0 0
\(226\) −5.79383 −0.385400
\(227\) 4.45231 0.295510 0.147755 0.989024i \(-0.452795\pi\)
0.147755 + 0.989024i \(0.452795\pi\)
\(228\) −8.11704 −0.537564
\(229\) −4.05249 −0.267796 −0.133898 0.990995i \(-0.542749\pi\)
−0.133898 + 0.990995i \(0.542749\pi\)
\(230\) 0 0
\(231\) −1.72928 −0.113778
\(232\) −2.00000 −0.131306
\(233\) 24.5327 1.60719 0.803595 0.595176i \(-0.202918\pi\)
0.803595 + 0.595176i \(0.202918\pi\)
\(234\) −5.52311 −0.361057
\(235\) 0 0
\(236\) −10.9817 −0.714846
\(237\) 3.79383 0.246436
\(238\) −7.04623 −0.456739
\(239\) 10.7755 0.697010 0.348505 0.937307i \(-0.386689\pi\)
0.348505 + 0.937307i \(0.386689\pi\)
\(240\) 0 0
\(241\) −10.7755 −0.694112 −0.347056 0.937844i \(-0.612819\pi\)
−0.347056 + 0.937844i \(0.612819\pi\)
\(242\) −10.2524 −0.659049
\(243\) −1.00000 −0.0641500
\(244\) −5.91087 −0.378405
\(245\) 0 0
\(246\) −3.52311 −0.224626
\(247\) −44.8313 −2.85255
\(248\) −3.25240 −0.206527
\(249\) 8.32320 0.527462
\(250\) 0 0
\(251\) −3.91087 −0.246852 −0.123426 0.992354i \(-0.539388\pi\)
−0.123426 + 0.992354i \(0.539388\pi\)
\(252\) −2.00000 −0.125988
\(253\) 0.864641 0.0543595
\(254\) −0.747604 −0.0469088
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) −28.0925 −1.75236 −0.876180 0.481985i \(-0.839916\pi\)
−0.876180 + 0.481985i \(0.839916\pi\)
\(258\) 1.13536 0.0706844
\(259\) 10.2707 0.638191
\(260\) 0 0
\(261\) −2.00000 −0.123797
\(262\) 19.7572 1.22060
\(263\) −29.5510 −1.82219 −0.911097 0.412192i \(-0.864763\pi\)
−0.911097 + 0.412192i \(0.864763\pi\)
\(264\) 0.864641 0.0532150
\(265\) 0 0
\(266\) −16.2341 −0.995375
\(267\) −4.77551 −0.292256
\(268\) 5.91087 0.361064
\(269\) −17.2803 −1.05360 −0.526799 0.849990i \(-0.676608\pi\)
−0.526799 + 0.849990i \(0.676608\pi\)
\(270\) 0 0
\(271\) 9.96336 0.605231 0.302615 0.953113i \(-0.402140\pi\)
0.302615 + 0.953113i \(0.402140\pi\)
\(272\) 3.52311 0.213620
\(273\) −11.0462 −0.668548
\(274\) −0.476886 −0.0288097
\(275\) 0 0
\(276\) 1.00000 0.0601929
\(277\) −27.0741 −1.62673 −0.813364 0.581756i \(-0.802366\pi\)
−0.813364 + 0.581756i \(0.802366\pi\)
\(278\) −3.45856 −0.207431
\(279\) −3.25240 −0.194716
\(280\) 0 0
\(281\) −27.2803 −1.62741 −0.813703 0.581281i \(-0.802552\pi\)
−0.813703 + 0.581281i \(0.802552\pi\)
\(282\) 5.52311 0.328897
\(283\) 30.9205 1.83803 0.919015 0.394222i \(-0.128986\pi\)
0.919015 + 0.394222i \(0.128986\pi\)
\(284\) −5.79383 −0.343801
\(285\) 0 0
\(286\) 4.77551 0.282382
\(287\) −7.04623 −0.415926
\(288\) 1.00000 0.0589256
\(289\) −4.58767 −0.269863
\(290\) 0 0
\(291\) −1.04623 −0.0613310
\(292\) −15.2524 −0.892579
\(293\) −11.0708 −0.646764 −0.323382 0.946269i \(-0.604820\pi\)
−0.323382 + 0.946269i \(0.604820\pi\)
\(294\) 3.00000 0.174964
\(295\) 0 0
\(296\) −5.13536 −0.298487
\(297\) 0.864641 0.0501716
\(298\) −8.86464 −0.513515
\(299\) 5.52311 0.319410
\(300\) 0 0
\(301\) 2.27072 0.130882
\(302\) −6.71096 −0.386172
\(303\) −9.04623 −0.519692
\(304\) 8.11704 0.465544
\(305\) 0 0
\(306\) 3.52311 0.201403
\(307\) −4.23407 −0.241651 −0.120826 0.992674i \(-0.538554\pi\)
−0.120826 + 0.992674i \(0.538554\pi\)
\(308\) 1.72928 0.0985350
\(309\) 11.3169 0.643799
\(310\) 0 0
\(311\) −13.2803 −0.753057 −0.376528 0.926405i \(-0.622882\pi\)
−0.376528 + 0.926405i \(0.622882\pi\)
\(312\) 5.52311 0.312685
\(313\) 31.9634 1.80668 0.903338 0.428930i \(-0.141109\pi\)
0.903338 + 0.428930i \(0.141109\pi\)
\(314\) 14.6864 0.828800
\(315\) 0 0
\(316\) −3.79383 −0.213420
\(317\) 13.2803 0.745896 0.372948 0.927852i \(-0.378347\pi\)
0.372948 + 0.927852i \(0.378347\pi\)
\(318\) 1.34153 0.0752291
\(319\) 1.72928 0.0968212
\(320\) 0 0
\(321\) 5.64015 0.314803
\(322\) 2.00000 0.111456
\(323\) 28.5972 1.59119
\(324\) 1.00000 0.0555556
\(325\) 0 0
\(326\) 23.2803 1.28938
\(327\) 12.1816 0.673643
\(328\) 3.52311 0.194531
\(329\) 11.0462 0.608998
\(330\) 0 0
\(331\) −11.8217 −0.649782 −0.324891 0.945752i \(-0.605328\pi\)
−0.324891 + 0.945752i \(0.605328\pi\)
\(332\) −8.32320 −0.456795
\(333\) −5.13536 −0.281416
\(334\) 5.93545 0.324773
\(335\) 0 0
\(336\) 2.00000 0.109109
\(337\) −13.2803 −0.723424 −0.361712 0.932290i \(-0.617808\pi\)
−0.361712 + 0.932290i \(0.617808\pi\)
\(338\) 17.5048 0.952135
\(339\) 5.79383 0.314678
\(340\) 0 0
\(341\) 2.81215 0.152287
\(342\) 8.11704 0.438919
\(343\) 20.0000 1.07990
\(344\) −1.13536 −0.0612145
\(345\) 0 0
\(346\) −25.8217 −1.38819
\(347\) 24.3632 1.30788 0.653942 0.756545i \(-0.273114\pi\)
0.653942 + 0.756545i \(0.273114\pi\)
\(348\) 2.00000 0.107211
\(349\) 3.49521 0.187094 0.0935471 0.995615i \(-0.470179\pi\)
0.0935471 + 0.995615i \(0.470179\pi\)
\(350\) 0 0
\(351\) 5.52311 0.294802
\(352\) −0.864641 −0.0460855
\(353\) −31.5789 −1.68078 −0.840388 0.541985i \(-0.817673\pi\)
−0.840388 + 0.541985i \(0.817673\pi\)
\(354\) 10.9817 0.583670
\(355\) 0 0
\(356\) 4.77551 0.253102
\(357\) 7.04623 0.372926
\(358\) −23.3449 −1.23381
\(359\) −21.1878 −1.11825 −0.559126 0.829083i \(-0.688863\pi\)
−0.559126 + 0.829083i \(0.688863\pi\)
\(360\) 0 0
\(361\) 46.8863 2.46770
\(362\) 13.3694 0.702682
\(363\) 10.2524 0.538111
\(364\) 11.0462 0.578980
\(365\) 0 0
\(366\) 5.91087 0.308966
\(367\) −13.0462 −0.681008 −0.340504 0.940243i \(-0.610598\pi\)
−0.340504 + 0.940243i \(0.610598\pi\)
\(368\) −1.00000 −0.0521286
\(369\) 3.52311 0.183406
\(370\) 0 0
\(371\) 2.68305 0.139297
\(372\) 3.25240 0.168629
\(373\) −18.8646 −0.976774 −0.488387 0.872627i \(-0.662415\pi\)
−0.488387 + 0.872627i \(0.662415\pi\)
\(374\) −3.04623 −0.157517
\(375\) 0 0
\(376\) −5.52311 −0.284833
\(377\) 11.0462 0.568910
\(378\) 2.00000 0.102869
\(379\) 26.7509 1.37410 0.687052 0.726609i \(-0.258905\pi\)
0.687052 + 0.726609i \(0.258905\pi\)
\(380\) 0 0
\(381\) 0.747604 0.0383009
\(382\) 12.2341 0.625950
\(383\) 22.5048 1.14994 0.574971 0.818174i \(-0.305013\pi\)
0.574971 + 0.818174i \(0.305013\pi\)
\(384\) −1.00000 −0.0510310
\(385\) 0 0
\(386\) −7.45856 −0.379631
\(387\) −1.13536 −0.0577135
\(388\) 1.04623 0.0531142
\(389\) −7.67680 −0.389229 −0.194614 0.980880i \(-0.562346\pi\)
−0.194614 + 0.980880i \(0.562346\pi\)
\(390\) 0 0
\(391\) −3.52311 −0.178172
\(392\) −3.00000 −0.151523
\(393\) −19.7572 −0.996618
\(394\) −2.00000 −0.100759
\(395\) 0 0
\(396\) −0.864641 −0.0434498
\(397\) 31.7938 1.59569 0.797843 0.602865i \(-0.205974\pi\)
0.797843 + 0.602865i \(0.205974\pi\)
\(398\) 22.8401 1.14487
\(399\) 16.2341 0.812720
\(400\) 0 0
\(401\) −18.0925 −0.903494 −0.451747 0.892146i \(-0.649199\pi\)
−0.451747 + 0.892146i \(0.649199\pi\)
\(402\) −5.91087 −0.294807
\(403\) 17.9634 0.894818
\(404\) 9.04623 0.450067
\(405\) 0 0
\(406\) 4.00000 0.198517
\(407\) 4.44024 0.220095
\(408\) −3.52311 −0.174420
\(409\) 2.00000 0.0988936 0.0494468 0.998777i \(-0.484254\pi\)
0.0494468 + 0.998777i \(0.484254\pi\)
\(410\) 0 0
\(411\) 0.476886 0.0235230
\(412\) −11.3169 −0.557546
\(413\) 21.9634 1.08075
\(414\) −1.00000 −0.0491473
\(415\) 0 0
\(416\) −5.52311 −0.270793
\(417\) 3.45856 0.169367
\(418\) −7.01832 −0.343277
\(419\) 20.4523 0.999161 0.499580 0.866268i \(-0.333488\pi\)
0.499580 + 0.866268i \(0.333488\pi\)
\(420\) 0 0
\(421\) −12.9571 −0.631490 −0.315745 0.948844i \(-0.602254\pi\)
−0.315745 + 0.948844i \(0.602254\pi\)
\(422\) 24.2341 1.17970
\(423\) −5.52311 −0.268543
\(424\) −1.34153 −0.0651503
\(425\) 0 0
\(426\) 5.79383 0.280712
\(427\) 11.8217 0.572094
\(428\) −5.64015 −0.272627
\(429\) −4.77551 −0.230564
\(430\) 0 0
\(431\) 33.3728 1.60751 0.803755 0.594961i \(-0.202832\pi\)
0.803755 + 0.594961i \(0.202832\pi\)
\(432\) −1.00000 −0.0481125
\(433\) −19.0096 −0.913542 −0.456771 0.889584i \(-0.650994\pi\)
−0.456771 + 0.889584i \(0.650994\pi\)
\(434\) 6.50479 0.312240
\(435\) 0 0
\(436\) −12.1816 −0.583392
\(437\) −8.11704 −0.388291
\(438\) 15.2524 0.728788
\(439\) 32.5972 1.55578 0.777891 0.628399i \(-0.216290\pi\)
0.777891 + 0.628399i \(0.216290\pi\)
\(440\) 0 0
\(441\) −3.00000 −0.142857
\(442\) −19.4586 −0.925549
\(443\) −4.00000 −0.190046 −0.0950229 0.995475i \(-0.530292\pi\)
−0.0950229 + 0.995475i \(0.530292\pi\)
\(444\) 5.13536 0.243713
\(445\) 0 0
\(446\) −13.9634 −0.661184
\(447\) 8.86464 0.419283
\(448\) −2.00000 −0.0944911
\(449\) 39.5789 1.86785 0.933923 0.357475i \(-0.116362\pi\)
0.933923 + 0.357475i \(0.116362\pi\)
\(450\) 0 0
\(451\) −3.04623 −0.143441
\(452\) −5.79383 −0.272519
\(453\) 6.71096 0.315308
\(454\) 4.45231 0.208957
\(455\) 0 0
\(456\) −8.11704 −0.380115
\(457\) −24.5048 −1.14629 −0.573143 0.819455i \(-0.694276\pi\)
−0.573143 + 0.819455i \(0.694276\pi\)
\(458\) −4.05249 −0.189360
\(459\) −3.52311 −0.164445
\(460\) 0 0
\(461\) 18.2341 0.849245 0.424623 0.905370i \(-0.360407\pi\)
0.424623 + 0.905370i \(0.360407\pi\)
\(462\) −1.72928 −0.0804535
\(463\) 4.20617 0.195477 0.0977386 0.995212i \(-0.468839\pi\)
0.0977386 + 0.995212i \(0.468839\pi\)
\(464\) −2.00000 −0.0928477
\(465\) 0 0
\(466\) 24.5327 1.13646
\(467\) −19.1912 −0.888062 −0.444031 0.896012i \(-0.646452\pi\)
−0.444031 + 0.896012i \(0.646452\pi\)
\(468\) −5.52311 −0.255306
\(469\) −11.8217 −0.545877
\(470\) 0 0
\(471\) −14.6864 −0.676713
\(472\) −10.9817 −0.505473
\(473\) 0.981678 0.0451376
\(474\) 3.79383 0.174257
\(475\) 0 0
\(476\) −7.04623 −0.322963
\(477\) −1.34153 −0.0614243
\(478\) 10.7755 0.492860
\(479\) −25.6801 −1.17335 −0.586677 0.809821i \(-0.699564\pi\)
−0.586677 + 0.809821i \(0.699564\pi\)
\(480\) 0 0
\(481\) 28.3632 1.29325
\(482\) −10.7755 −0.490811
\(483\) −2.00000 −0.0910032
\(484\) −10.2524 −0.466018
\(485\) 0 0
\(486\) −1.00000 −0.0453609
\(487\) −8.74760 −0.396392 −0.198196 0.980162i \(-0.563508\pi\)
−0.198196 + 0.980162i \(0.563508\pi\)
\(488\) −5.91087 −0.267572
\(489\) −23.2803 −1.05277
\(490\) 0 0
\(491\) 23.8863 1.07797 0.538987 0.842314i \(-0.318807\pi\)
0.538987 + 0.842314i \(0.318807\pi\)
\(492\) −3.52311 −0.158834
\(493\) −7.04623 −0.317346
\(494\) −44.8313 −2.01706
\(495\) 0 0
\(496\) −3.25240 −0.146037
\(497\) 11.5877 0.519778
\(498\) 8.32320 0.372972
\(499\) 29.7293 1.33087 0.665433 0.746458i \(-0.268247\pi\)
0.665433 + 0.746458i \(0.268247\pi\)
\(500\) 0 0
\(501\) −5.93545 −0.265176
\(502\) −3.91087 −0.174551
\(503\) −0.775511 −0.0345783 −0.0172892 0.999851i \(-0.505504\pi\)
−0.0172892 + 0.999851i \(0.505504\pi\)
\(504\) −2.00000 −0.0890871
\(505\) 0 0
\(506\) 0.864641 0.0384380
\(507\) −17.5048 −0.777415
\(508\) −0.747604 −0.0331696
\(509\) −26.0000 −1.15243 −0.576215 0.817298i \(-0.695471\pi\)
−0.576215 + 0.817298i \(0.695471\pi\)
\(510\) 0 0
\(511\) 30.5048 1.34945
\(512\) 1.00000 0.0441942
\(513\) −8.11704 −0.358376
\(514\) −28.0925 −1.23911
\(515\) 0 0
\(516\) 1.13536 0.0499814
\(517\) 4.77551 0.210027
\(518\) 10.2707 0.451269
\(519\) 25.8217 1.13345
\(520\) 0 0
\(521\) 13.4219 0.588025 0.294012 0.955802i \(-0.405009\pi\)
0.294012 + 0.955802i \(0.405009\pi\)
\(522\) −2.00000 −0.0875376
\(523\) 41.9667 1.83507 0.917537 0.397649i \(-0.130174\pi\)
0.917537 + 0.397649i \(0.130174\pi\)
\(524\) 19.7572 0.863097
\(525\) 0 0
\(526\) −29.5510 −1.28849
\(527\) −11.4586 −0.499143
\(528\) 0.864641 0.0376287
\(529\) 1.00000 0.0434783
\(530\) 0 0
\(531\) −10.9817 −0.476564
\(532\) −16.2341 −0.703836
\(533\) −19.4586 −0.842844
\(534\) −4.77551 −0.206657
\(535\) 0 0
\(536\) 5.91087 0.255311
\(537\) 23.3449 1.00740
\(538\) −17.2803 −0.745007
\(539\) 2.59392 0.111728
\(540\) 0 0
\(541\) −1.58767 −0.0682591 −0.0341295 0.999417i \(-0.510866\pi\)
−0.0341295 + 0.999417i \(0.510866\pi\)
\(542\) 9.96336 0.427963
\(543\) −13.3694 −0.573737
\(544\) 3.52311 0.151052
\(545\) 0 0
\(546\) −11.0462 −0.472735
\(547\) 20.6464 0.882777 0.441388 0.897316i \(-0.354486\pi\)
0.441388 + 0.897316i \(0.354486\pi\)
\(548\) −0.476886 −0.0203715
\(549\) −5.91087 −0.252270
\(550\) 0 0
\(551\) −16.2341 −0.691595
\(552\) 1.00000 0.0425628
\(553\) 7.58767 0.322660
\(554\) −27.0741 −1.15027
\(555\) 0 0
\(556\) −3.45856 −0.146676
\(557\) −9.47063 −0.401283 −0.200642 0.979665i \(-0.564303\pi\)
−0.200642 + 0.979665i \(0.564303\pi\)
\(558\) −3.25240 −0.137685
\(559\) 6.27072 0.265223
\(560\) 0 0
\(561\) 3.04623 0.128612
\(562\) −27.2803 −1.15075
\(563\) −1.51105 −0.0636832 −0.0318416 0.999493i \(-0.510137\pi\)
−0.0318416 + 0.999493i \(0.510137\pi\)
\(564\) 5.52311 0.232565
\(565\) 0 0
\(566\) 30.9205 1.29968
\(567\) −2.00000 −0.0839921
\(568\) −5.79383 −0.243104
\(569\) −20.0000 −0.838444 −0.419222 0.907884i \(-0.637697\pi\)
−0.419222 + 0.907884i \(0.637697\pi\)
\(570\) 0 0
\(571\) −35.1633 −1.47154 −0.735768 0.677233i \(-0.763179\pi\)
−0.735768 + 0.677233i \(0.763179\pi\)
\(572\) 4.77551 0.199674
\(573\) −12.2341 −0.511086
\(574\) −7.04623 −0.294104
\(575\) 0 0
\(576\) 1.00000 0.0416667
\(577\) 22.7110 0.945470 0.472735 0.881205i \(-0.343267\pi\)
0.472735 + 0.881205i \(0.343267\pi\)
\(578\) −4.58767 −0.190822
\(579\) 7.45856 0.309967
\(580\) 0 0
\(581\) 16.6464 0.690609
\(582\) −1.04623 −0.0433676
\(583\) 1.15994 0.0480398
\(584\) −15.2524 −0.631149
\(585\) 0 0
\(586\) −11.0708 −0.457331
\(587\) −3.76593 −0.155436 −0.0777182 0.996975i \(-0.524763\pi\)
−0.0777182 + 0.996975i \(0.524763\pi\)
\(588\) 3.00000 0.123718
\(589\) −26.3998 −1.08779
\(590\) 0 0
\(591\) 2.00000 0.0822690
\(592\) −5.13536 −0.211062
\(593\) −1.04623 −0.0429635 −0.0214817 0.999769i \(-0.506838\pi\)
−0.0214817 + 0.999769i \(0.506838\pi\)
\(594\) 0.864641 0.0354766
\(595\) 0 0
\(596\) −8.86464 −0.363110
\(597\) −22.8401 −0.934781
\(598\) 5.52311 0.225857
\(599\) −21.2803 −0.869490 −0.434745 0.900554i \(-0.643161\pi\)
−0.434745 + 0.900554i \(0.643161\pi\)
\(600\) 0 0
\(601\) 30.8034 1.25650 0.628249 0.778012i \(-0.283772\pi\)
0.628249 + 0.778012i \(0.283772\pi\)
\(602\) 2.27072 0.0925476
\(603\) 5.91087 0.240709
\(604\) −6.71096 −0.273065
\(605\) 0 0
\(606\) −9.04623 −0.367478
\(607\) 22.0925 0.896705 0.448353 0.893857i \(-0.352011\pi\)
0.448353 + 0.893857i \(0.352011\pi\)
\(608\) 8.11704 0.329189
\(609\) −4.00000 −0.162088
\(610\) 0 0
\(611\) 30.5048 1.23409
\(612\) 3.52311 0.142413
\(613\) 41.4985 1.67611 0.838055 0.545586i \(-0.183693\pi\)
0.838055 + 0.545586i \(0.183693\pi\)
\(614\) −4.23407 −0.170873
\(615\) 0 0
\(616\) 1.72928 0.0696747
\(617\) 30.9325 1.24530 0.622648 0.782502i \(-0.286057\pi\)
0.622648 + 0.782502i \(0.286057\pi\)
\(618\) 11.3169 0.455234
\(619\) 14.5660 0.585458 0.292729 0.956196i \(-0.405437\pi\)
0.292729 + 0.956196i \(0.405437\pi\)
\(620\) 0 0
\(621\) 1.00000 0.0401286
\(622\) −13.2803 −0.532492
\(623\) −9.55102 −0.382654
\(624\) 5.52311 0.221102
\(625\) 0 0
\(626\) 31.9634 1.27751
\(627\) 7.01832 0.280285
\(628\) 14.6864 0.586050
\(629\) −18.0925 −0.721394
\(630\) 0 0
\(631\) −38.4036 −1.52882 −0.764412 0.644729i \(-0.776970\pi\)
−0.764412 + 0.644729i \(0.776970\pi\)
\(632\) −3.79383 −0.150911
\(633\) −24.2341 −0.963218
\(634\) 13.2803 0.527428
\(635\) 0 0
\(636\) 1.34153 0.0531950
\(637\) 16.5693 0.656501
\(638\) 1.72928 0.0684629
\(639\) −5.79383 −0.229200
\(640\) 0 0
\(641\) −31.9267 −1.26103 −0.630515 0.776177i \(-0.717156\pi\)
−0.630515 + 0.776177i \(0.717156\pi\)
\(642\) 5.64015 0.222599
\(643\) 4.72302 0.186258 0.0931289 0.995654i \(-0.470313\pi\)
0.0931289 + 0.995654i \(0.470313\pi\)
\(644\) 2.00000 0.0788110
\(645\) 0 0
\(646\) 28.5972 1.12514
\(647\) 43.4586 1.70853 0.854266 0.519836i \(-0.174007\pi\)
0.854266 + 0.519836i \(0.174007\pi\)
\(648\) 1.00000 0.0392837
\(649\) 9.49521 0.372720
\(650\) 0 0
\(651\) −6.50479 −0.254943
\(652\) 23.2803 0.911727
\(653\) −2.36318 −0.0924782 −0.0462391 0.998930i \(-0.514724\pi\)
−0.0462391 + 0.998930i \(0.514724\pi\)
\(654\) 12.1816 0.476338
\(655\) 0 0
\(656\) 3.52311 0.137555
\(657\) −15.2524 −0.595053
\(658\) 11.0462 0.430627
\(659\) −22.4157 −0.873190 −0.436595 0.899658i \(-0.643816\pi\)
−0.436595 + 0.899658i \(0.643816\pi\)
\(660\) 0 0
\(661\) −14.9937 −0.583189 −0.291594 0.956542i \(-0.594186\pi\)
−0.291594 + 0.956542i \(0.594186\pi\)
\(662\) −11.8217 −0.459465
\(663\) 19.4586 0.755708
\(664\) −8.32320 −0.323003
\(665\) 0 0
\(666\) −5.13536 −0.198991
\(667\) 2.00000 0.0774403
\(668\) 5.93545 0.229649
\(669\) 13.9634 0.539855
\(670\) 0 0
\(671\) 5.11078 0.197299
\(672\) 2.00000 0.0771517
\(673\) 49.2158 1.89713 0.948564 0.316586i \(-0.102536\pi\)
0.948564 + 0.316586i \(0.102536\pi\)
\(674\) −13.2803 −0.511538
\(675\) 0 0
\(676\) 17.5048 0.673261
\(677\) 5.34153 0.205292 0.102646 0.994718i \(-0.467269\pi\)
0.102646 + 0.994718i \(0.467269\pi\)
\(678\) 5.79383 0.222511
\(679\) −2.09246 −0.0803011
\(680\) 0 0
\(681\) −4.45231 −0.170613
\(682\) 2.81215 0.107683
\(683\) −27.1512 −1.03891 −0.519456 0.854497i \(-0.673865\pi\)
−0.519456 + 0.854497i \(0.673865\pi\)
\(684\) 8.11704 0.310363
\(685\) 0 0
\(686\) 20.0000 0.763604
\(687\) 4.05249 0.154612
\(688\) −1.13536 −0.0432852
\(689\) 7.40940 0.282276
\(690\) 0 0
\(691\) 12.1050 0.460495 0.230247 0.973132i \(-0.426046\pi\)
0.230247 + 0.973132i \(0.426046\pi\)
\(692\) −25.8217 −0.981595
\(693\) 1.72928 0.0656900
\(694\) 24.3632 0.924814
\(695\) 0 0
\(696\) 2.00000 0.0758098
\(697\) 12.4123 0.470151
\(698\) 3.49521 0.132296
\(699\) −24.5327 −0.927912
\(700\) 0 0
\(701\) 2.59392 0.0979711 0.0489856 0.998799i \(-0.484401\pi\)
0.0489856 + 0.998799i \(0.484401\pi\)
\(702\) 5.52311 0.208457
\(703\) −41.6839 −1.57214
\(704\) −0.864641 −0.0325874
\(705\) 0 0
\(706\) −31.5789 −1.18849
\(707\) −18.0925 −0.680437
\(708\) 10.9817 0.412717
\(709\) 4.00333 0.150348 0.0751741 0.997170i \(-0.476049\pi\)
0.0751741 + 0.997170i \(0.476049\pi\)
\(710\) 0 0
\(711\) −3.79383 −0.142280
\(712\) 4.77551 0.178970
\(713\) 3.25240 0.121803
\(714\) 7.04623 0.263698
\(715\) 0 0
\(716\) −23.3449 −0.872438
\(717\) −10.7755 −0.402419
\(718\) −21.1878 −0.790723
\(719\) 20.2707 0.755970 0.377985 0.925812i \(-0.376617\pi\)
0.377985 + 0.925812i \(0.376617\pi\)
\(720\) 0 0
\(721\) 22.6339 0.842930
\(722\) 46.8863 1.74493
\(723\) 10.7755 0.400746
\(724\) 13.3694 0.496871
\(725\) 0 0
\(726\) 10.2524 0.380502
\(727\) 32.6339 1.21032 0.605162 0.796102i \(-0.293108\pi\)
0.605162 + 0.796102i \(0.293108\pi\)
\(728\) 11.0462 0.409400
\(729\) 1.00000 0.0370370
\(730\) 0 0
\(731\) −4.00000 −0.147945
\(732\) 5.91087 0.218472
\(733\) 1.13536 0.0419354 0.0209677 0.999780i \(-0.493325\pi\)
0.0209677 + 0.999780i \(0.493325\pi\)
\(734\) −13.0462 −0.481545
\(735\) 0 0
\(736\) −1.00000 −0.0368605
\(737\) −5.11078 −0.188258
\(738\) 3.52311 0.129688
\(739\) −29.4219 −1.08230 −0.541151 0.840925i \(-0.682011\pi\)
−0.541151 + 0.840925i \(0.682011\pi\)
\(740\) 0 0
\(741\) 44.8313 1.64692
\(742\) 2.68305 0.0984980
\(743\) 27.4586 1.00736 0.503678 0.863891i \(-0.331980\pi\)
0.503678 + 0.863891i \(0.331980\pi\)
\(744\) 3.25240 0.119239
\(745\) 0 0
\(746\) −18.8646 −0.690684
\(747\) −8.32320 −0.304530
\(748\) −3.04623 −0.111381
\(749\) 11.2803 0.412173
\(750\) 0 0
\(751\) 4.02791 0.146980 0.0734902 0.997296i \(-0.476586\pi\)
0.0734902 + 0.997296i \(0.476586\pi\)
\(752\) −5.52311 −0.201407
\(753\) 3.91087 0.142520
\(754\) 11.0462 0.402280
\(755\) 0 0
\(756\) 2.00000 0.0727393
\(757\) 31.5110 1.14529 0.572644 0.819804i \(-0.305918\pi\)
0.572644 + 0.819804i \(0.305918\pi\)
\(758\) 26.7509 0.971638
\(759\) −0.864641 −0.0313845
\(760\) 0 0
\(761\) −19.9634 −0.723671 −0.361836 0.932242i \(-0.617850\pi\)
−0.361836 + 0.932242i \(0.617850\pi\)
\(762\) 0.747604 0.0270828
\(763\) 24.3632 0.882006
\(764\) 12.2341 0.442613
\(765\) 0 0
\(766\) 22.5048 0.813131
\(767\) 60.6531 2.19006
\(768\) −1.00000 −0.0360844
\(769\) −46.4190 −1.67391 −0.836956 0.547271i \(-0.815667\pi\)
−0.836956 + 0.547271i \(0.815667\pi\)
\(770\) 0 0
\(771\) 28.0925 1.01173
\(772\) −7.45856 −0.268440
\(773\) 29.0342 1.04429 0.522143 0.852858i \(-0.325133\pi\)
0.522143 + 0.852858i \(0.325133\pi\)
\(774\) −1.13536 −0.0408096
\(775\) 0 0
\(776\) 1.04623 0.0375574
\(777\) −10.2707 −0.368460
\(778\) −7.67680 −0.275226
\(779\) 28.5972 1.02460
\(780\) 0 0
\(781\) 5.00958 0.179257
\(782\) −3.52311 −0.125986
\(783\) 2.00000 0.0714742
\(784\) −3.00000 −0.107143
\(785\) 0 0
\(786\) −19.7572 −0.704716
\(787\) −35.7693 −1.27504 −0.637518 0.770435i \(-0.720039\pi\)
−0.637518 + 0.770435i \(0.720039\pi\)
\(788\) −2.00000 −0.0712470
\(789\) 29.5510 1.05204
\(790\) 0 0
\(791\) 11.5877 0.412010
\(792\) −0.864641 −0.0307237
\(793\) 32.6464 1.15931
\(794\) 31.7938 1.12832
\(795\) 0 0
\(796\) 22.8401 0.809545
\(797\) 22.8367 0.808919 0.404459 0.914556i \(-0.367460\pi\)
0.404459 + 0.914556i \(0.367460\pi\)
\(798\) 16.2341 0.574680
\(799\) −19.4586 −0.688394
\(800\) 0 0
\(801\) 4.77551 0.168734
\(802\) −18.0925 −0.638867
\(803\) 13.1878 0.465389
\(804\) −5.91087 −0.208460
\(805\) 0 0
\(806\) 17.9634 0.632732
\(807\) 17.2803 0.608295
\(808\) 9.04623 0.318245
\(809\) −36.9171 −1.29794 −0.648969 0.760815i \(-0.724799\pi\)
−0.648969 + 0.760815i \(0.724799\pi\)
\(810\) 0 0
\(811\) 51.8217 1.81971 0.909854 0.414929i \(-0.136194\pi\)
0.909854 + 0.414929i \(0.136194\pi\)
\(812\) 4.00000 0.140372
\(813\) −9.96336 −0.349430
\(814\) 4.44024 0.155630
\(815\) 0 0
\(816\) −3.52311 −0.123334
\(817\) −9.21575 −0.322418
\(818\) 2.00000 0.0699284
\(819\) 11.0462 0.385986
\(820\) 0 0
\(821\) −17.8709 −0.623699 −0.311849 0.950132i \(-0.600948\pi\)
−0.311849 + 0.950132i \(0.600948\pi\)
\(822\) 0.476886 0.0166333
\(823\) −9.00958 −0.314054 −0.157027 0.987594i \(-0.550191\pi\)
−0.157027 + 0.987594i \(0.550191\pi\)
\(824\) −11.3169 −0.394245
\(825\) 0 0
\(826\) 21.9634 0.764203
\(827\) −42.1083 −1.46425 −0.732125 0.681171i \(-0.761471\pi\)
−0.732125 + 0.681171i \(0.761471\pi\)
\(828\) −1.00000 −0.0347524
\(829\) −35.5510 −1.23474 −0.617369 0.786674i \(-0.711801\pi\)
−0.617369 + 0.786674i \(0.711801\pi\)
\(830\) 0 0
\(831\) 27.0741 0.939191
\(832\) −5.52311 −0.191480
\(833\) −10.5693 −0.366206
\(834\) 3.45856 0.119760
\(835\) 0 0
\(836\) −7.01832 −0.242734
\(837\) 3.25240 0.112419
\(838\) 20.4523 0.706513
\(839\) −25.6801 −0.886576 −0.443288 0.896379i \(-0.646188\pi\)
−0.443288 + 0.896379i \(0.646188\pi\)
\(840\) 0 0
\(841\) −25.0000 −0.862069
\(842\) −12.9571 −0.446531
\(843\) 27.2803 0.939584
\(844\) 24.2341 0.834171
\(845\) 0 0
\(846\) −5.52311 −0.189889
\(847\) 20.5048 0.704553
\(848\) −1.34153 −0.0460682
\(849\) −30.9205 −1.06119
\(850\) 0 0
\(851\) 5.13536 0.176038
\(852\) 5.79383 0.198493
\(853\) 11.6647 0.399393 0.199696 0.979858i \(-0.436004\pi\)
0.199696 + 0.979858i \(0.436004\pi\)
\(854\) 11.8217 0.404532
\(855\) 0 0
\(856\) −5.64015 −0.192776
\(857\) −41.5423 −1.41906 −0.709529 0.704677i \(-0.751092\pi\)
−0.709529 + 0.704677i \(0.751092\pi\)
\(858\) −4.77551 −0.163033
\(859\) 21.5510 0.735311 0.367656 0.929962i \(-0.380161\pi\)
0.367656 + 0.929962i \(0.380161\pi\)
\(860\) 0 0
\(861\) 7.04623 0.240135
\(862\) 33.3728 1.13668
\(863\) 6.91713 0.235462 0.117731 0.993046i \(-0.462438\pi\)
0.117731 + 0.993046i \(0.462438\pi\)
\(864\) −1.00000 −0.0340207
\(865\) 0 0
\(866\) −19.0096 −0.645972
\(867\) 4.58767 0.155805
\(868\) 6.50479 0.220787
\(869\) 3.28030 0.111277
\(870\) 0 0
\(871\) −32.6464 −1.10618
\(872\) −12.1816 −0.412521
\(873\) 1.04623 0.0354095
\(874\) −8.11704 −0.274563
\(875\) 0 0
\(876\) 15.2524 0.515331
\(877\) −2.53270 −0.0855232 −0.0427616 0.999085i \(-0.513616\pi\)
−0.0427616 + 0.999085i \(0.513616\pi\)
\(878\) 32.5972 1.10010
\(879\) 11.0708 0.373409
\(880\) 0 0
\(881\) −10.8680 −0.366151 −0.183076 0.983099i \(-0.558605\pi\)
−0.183076 + 0.983099i \(0.558605\pi\)
\(882\) −3.00000 −0.101015
\(883\) −40.4190 −1.36021 −0.680104 0.733116i \(-0.738065\pi\)
−0.680104 + 0.733116i \(0.738065\pi\)
\(884\) −19.4586 −0.654462
\(885\) 0 0
\(886\) −4.00000 −0.134383
\(887\) 7.48647 0.251371 0.125686 0.992070i \(-0.459887\pi\)
0.125686 + 0.992070i \(0.459887\pi\)
\(888\) 5.13536 0.172331
\(889\) 1.49521 0.0501477
\(890\) 0 0
\(891\) −0.864641 −0.0289666
\(892\) −13.9634 −0.467528
\(893\) −44.8313 −1.50022
\(894\) 8.86464 0.296478
\(895\) 0 0
\(896\) −2.00000 −0.0668153
\(897\) −5.52311 −0.184411
\(898\) 39.5789 1.32077
\(899\) 6.50479 0.216947
\(900\) 0 0
\(901\) −4.72635 −0.157458
\(902\) −3.04623 −0.101428
\(903\) −2.27072 −0.0755648
\(904\) −5.79383 −0.192700
\(905\) 0 0
\(906\) 6.71096 0.222957
\(907\) 22.6864 0.753289 0.376644 0.926358i \(-0.377078\pi\)
0.376644 + 0.926358i \(0.377078\pi\)
\(908\) 4.45231 0.147755
\(909\) 9.04623 0.300044
\(910\) 0 0
\(911\) −35.8217 −1.18683 −0.593414 0.804898i \(-0.702220\pi\)
−0.593414 + 0.804898i \(0.702220\pi\)
\(912\) −8.11704 −0.268782
\(913\) 7.19658 0.238172
\(914\) −24.5048 −0.810546
\(915\) 0 0
\(916\) −4.05249 −0.133898
\(917\) −39.5144 −1.30488
\(918\) −3.52311 −0.116280
\(919\) −13.2890 −0.438365 −0.219182 0.975684i \(-0.570339\pi\)
−0.219182 + 0.975684i \(0.570339\pi\)
\(920\) 0 0
\(921\) 4.23407 0.139517
\(922\) 18.2341 0.600507
\(923\) 32.0000 1.05329
\(924\) −1.72928 −0.0568892
\(925\) 0 0
\(926\) 4.20617 0.138223
\(927\) −11.3169 −0.371697
\(928\) −2.00000 −0.0656532
\(929\) 41.1299 1.34943 0.674715 0.738078i \(-0.264267\pi\)
0.674715 + 0.738078i \(0.264267\pi\)
\(930\) 0 0
\(931\) −24.3511 −0.798075
\(932\) 24.5327 0.803595
\(933\) 13.2803 0.434778
\(934\) −19.1912 −0.627954
\(935\) 0 0
\(936\) −5.52311 −0.180529
\(937\) −22.7755 −0.744043 −0.372022 0.928224i \(-0.621335\pi\)
−0.372022 + 0.928224i \(0.621335\pi\)
\(938\) −11.8217 −0.385993
\(939\) −31.9634 −1.04308
\(940\) 0 0
\(941\) −54.7788 −1.78574 −0.892870 0.450315i \(-0.851312\pi\)
−0.892870 + 0.450315i \(0.851312\pi\)
\(942\) −14.6864 −0.478508
\(943\) −3.52311 −0.114728
\(944\) −10.9817 −0.357423
\(945\) 0 0
\(946\) 0.981678 0.0319171
\(947\) 11.4094 0.370756 0.185378 0.982667i \(-0.440649\pi\)
0.185378 + 0.982667i \(0.440649\pi\)
\(948\) 3.79383 0.123218
\(949\) 84.2407 2.73457
\(950\) 0 0
\(951\) −13.2803 −0.430643
\(952\) −7.04623 −0.228370
\(953\) 41.6714 1.34987 0.674934 0.737878i \(-0.264172\pi\)
0.674934 + 0.737878i \(0.264172\pi\)
\(954\) −1.34153 −0.0434335
\(955\) 0 0
\(956\) 10.7755 0.348505
\(957\) −1.72928 −0.0558997
\(958\) −25.6801 −0.829687
\(959\) 0.953771 0.0307989
\(960\) 0 0
\(961\) −20.4219 −0.658772
\(962\) 28.3632 0.914465
\(963\) −5.64015 −0.181751
\(964\) −10.7755 −0.347056
\(965\) 0 0
\(966\) −2.00000 −0.0643489
\(967\) 3.17533 0.102112 0.0510559 0.998696i \(-0.483741\pi\)
0.0510559 + 0.998696i \(0.483741\pi\)
\(968\) −10.2524 −0.329524
\(969\) −28.5972 −0.918676
\(970\) 0 0
\(971\) −35.7326 −1.14671 −0.573357 0.819306i \(-0.694359\pi\)
−0.573357 + 0.819306i \(0.694359\pi\)
\(972\) −1.00000 −0.0320750
\(973\) 6.91713 0.221753
\(974\) −8.74760 −0.280291
\(975\) 0 0
\(976\) −5.91087 −0.189202
\(977\) 2.02791 0.0648785 0.0324392 0.999474i \(-0.489672\pi\)
0.0324392 + 0.999474i \(0.489672\pi\)
\(978\) −23.2803 −0.744422
\(979\) −4.12910 −0.131967
\(980\) 0 0
\(981\) −12.1816 −0.388928
\(982\) 23.8863 0.762242
\(983\) −1.67347 −0.0533754 −0.0266877 0.999644i \(-0.508496\pi\)
−0.0266877 + 0.999644i \(0.508496\pi\)
\(984\) −3.52311 −0.112313
\(985\) 0 0
\(986\) −7.04623 −0.224398
\(987\) −11.0462 −0.351605
\(988\) −44.8313 −1.42627
\(989\) 1.13536 0.0361023
\(990\) 0 0
\(991\) −55.8496 −1.77412 −0.887061 0.461652i \(-0.847257\pi\)
−0.887061 + 0.461652i \(0.847257\pi\)
\(992\) −3.25240 −0.103264
\(993\) 11.8217 0.375152
\(994\) 11.5877 0.367538
\(995\) 0 0
\(996\) 8.32320 0.263731
\(997\) 44.2062 1.40002 0.700012 0.714131i \(-0.253178\pi\)
0.700012 + 0.714131i \(0.253178\pi\)
\(998\) 29.7293 0.941064
\(999\) 5.13536 0.162476
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 3450.2.a.br.1.2 3
5.2 odd 4 690.2.d.d.139.5 yes 6
5.3 odd 4 690.2.d.d.139.2 6
5.4 even 2 3450.2.a.bq.1.2 3
15.2 even 4 2070.2.d.d.829.2 6
15.8 even 4 2070.2.d.d.829.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
690.2.d.d.139.2 6 5.3 odd 4
690.2.d.d.139.5 yes 6 5.2 odd 4
2070.2.d.d.829.2 6 15.2 even 4
2070.2.d.d.829.5 6 15.8 even 4
3450.2.a.bq.1.2 3 5.4 even 2
3450.2.a.br.1.2 3 1.1 even 1 trivial