Properties

Label 3330.2.a.bk.1.3
Level $3330$
Weight $2$
Character 3330.1
Self dual yes
Analytic conductor $26.590$
Analytic rank $0$
Dimension $5$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 3330 = 2 \cdot 3^{2} \cdot 5 \cdot 37 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 3330.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(26.5901838731\)
Analytic rank: \(0\)
Dimension: \(5\)
Coefficient field: 5.5.23544108.1
Defining polynomial: \(x^{5} - x^{4} - 20 x^{3} + 39 x^{2} + 9 x - 36\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(-0.892002\) of defining polynomial
Character \(\chi\) \(=\) 3330.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.647226 q^{7} -1.00000 q^{8} +O(q^{10})\) \(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{5} +0.647226 q^{7} -1.00000 q^{8} +1.00000 q^{10} -3.68293 q^{11} -3.22455 q^{13} -0.647226 q^{14} +1.00000 q^{16} -1.35277 q^{17} +6.30301 q^{19} -1.00000 q^{20} +3.68293 q^{22} -5.22455 q^{23} +1.00000 q^{25} +3.22455 q^{26} +0.647226 q^{28} +3.87178 q^{29} -5.98594 q^{31} -1.00000 q^{32} +1.35277 q^{34} -0.647226 q^{35} +1.00000 q^{37} -6.30301 q^{38} +1.00000 q^{40} +0.242382 q^{41} +1.75762 q^{43} -3.68293 q^{44} +5.22455 q^{46} +6.44432 q^{47} -6.58110 q^{49} -1.00000 q^{50} -3.22455 q^{52} -5.46694 q^{53} +3.68293 q^{55} -0.647226 q^{56} -3.87178 q^{58} +7.36586 q^{59} +14.4755 q^{61} +5.98594 q^{62} +1.00000 q^{64} +3.22455 q^{65} +6.07141 q^{67} -1.35277 q^{68} +0.647226 q^{70} -8.22832 q^{71} +8.08702 q^{73} -1.00000 q^{74} +6.30301 q^{76} -2.38369 q^{77} +10.2283 q^{79} -1.00000 q^{80} -0.242382 q^{82} +2.14131 q^{83} +1.35277 q^{85} -1.75762 q^{86} +3.68293 q^{88} +5.81346 q^{89} -2.08702 q^{91} -5.22455 q^{92} -6.44432 q^{94} -6.30301 q^{95} -4.20194 q^{97} +6.58110 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 5q - 5q^{2} + 5q^{4} - 5q^{5} + 3q^{7} - 5q^{8} + O(q^{10}) \) \( 5q - 5q^{2} + 5q^{4} - 5q^{5} + 3q^{7} - 5q^{8} + 5q^{10} - 5q^{11} + 5q^{13} - 3q^{14} + 5q^{16} - 7q^{17} - q^{19} - 5q^{20} + 5q^{22} - 5q^{23} + 5q^{25} - 5q^{26} + 3q^{28} - 2q^{29} + 16q^{31} - 5q^{32} + 7q^{34} - 3q^{35} + 5q^{37} + q^{38} + 5q^{40} - 2q^{41} + 12q^{43} - 5q^{44} + 5q^{46} - 6q^{47} + 16q^{49} - 5q^{50} + 5q^{52} - 3q^{53} + 5q^{55} - 3q^{56} + 2q^{58} + 10q^{59} + 16q^{61} - 16q^{62} + 5q^{64} - 5q^{65} + 4q^{67} - 7q^{68} + 3q^{70} + 8q^{71} - 3q^{73} - 5q^{74} - q^{76} - 3q^{77} + 2q^{79} - 5q^{80} + 2q^{82} + 5q^{83} + 7q^{85} - 12q^{86} + 5q^{88} + 7q^{89} + 33q^{91} - 5q^{92} + 6q^{94} + q^{95} + 14q^{97} - 16q^{98} + O(q^{100}) \)

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
\(7\) 0.647226 0.244629 0.122314 0.992491i \(-0.460968\pi\)
0.122314 + 0.992491i \(0.460968\pi\)
\(8\) −1.00000 −0.353553
\(9\) 0 0
\(10\) 1.00000 0.316228
\(11\) −3.68293 −1.11045 −0.555223 0.831702i \(-0.687367\pi\)
−0.555223 + 0.831702i \(0.687367\pi\)
\(12\) 0 0
\(13\) −3.22455 −0.894330 −0.447165 0.894451i \(-0.647566\pi\)
−0.447165 + 0.894451i \(0.647566\pi\)
\(14\) −0.647226 −0.172979
\(15\) 0 0
\(16\) 1.00000 0.250000
\(17\) −1.35277 −0.328096 −0.164048 0.986452i \(-0.552455\pi\)
−0.164048 + 0.986452i \(0.552455\pi\)
\(18\) 0 0
\(19\) 6.30301 1.44601 0.723005 0.690843i \(-0.242760\pi\)
0.723005 + 0.690843i \(0.242760\pi\)
\(20\) −1.00000 −0.223607
\(21\) 0 0
\(22\) 3.68293 0.785204
\(23\) −5.22455 −1.08939 −0.544697 0.838633i \(-0.683355\pi\)
−0.544697 + 0.838633i \(0.683355\pi\)
\(24\) 0 0
\(25\) 1.00000 0.200000
\(26\) 3.22455 0.632387
\(27\) 0 0
\(28\) 0.647226 0.122314
\(29\) 3.87178 0.718972 0.359486 0.933151i \(-0.382952\pi\)
0.359486 + 0.933151i \(0.382952\pi\)
\(30\) 0 0
\(31\) −5.98594 −1.07511 −0.537553 0.843230i \(-0.680651\pi\)
−0.537553 + 0.843230i \(0.680651\pi\)
\(32\) −1.00000 −0.176777
\(33\) 0 0
\(34\) 1.35277 0.231999
\(35\) −0.647226 −0.109401
\(36\) 0 0
\(37\) 1.00000 0.164399
\(38\) −6.30301 −1.02248
\(39\) 0 0
\(40\) 1.00000 0.158114
\(41\) 0.242382 0.0378537 0.0189269 0.999821i \(-0.493975\pi\)
0.0189269 + 0.999821i \(0.493975\pi\)
\(42\) 0 0
\(43\) 1.75762 0.268034 0.134017 0.990979i \(-0.457212\pi\)
0.134017 + 0.990979i \(0.457212\pi\)
\(44\) −3.68293 −0.555223
\(45\) 0 0
\(46\) 5.22455 0.770318
\(47\) 6.44432 0.940001 0.470000 0.882666i \(-0.344254\pi\)
0.470000 + 0.882666i \(0.344254\pi\)
\(48\) 0 0
\(49\) −6.58110 −0.940157
\(50\) −1.00000 −0.141421
\(51\) 0 0
\(52\) −3.22455 −0.447165
\(53\) −5.46694 −0.750941 −0.375471 0.926834i \(-0.622519\pi\)
−0.375471 + 0.926834i \(0.622519\pi\)
\(54\) 0 0
\(55\) 3.68293 0.496606
\(56\) −0.647226 −0.0864893
\(57\) 0 0
\(58\) −3.87178 −0.508390
\(59\) 7.36586 0.958954 0.479477 0.877555i \(-0.340826\pi\)
0.479477 + 0.877555i \(0.340826\pi\)
\(60\) 0 0
\(61\) 14.4755 1.85340 0.926699 0.375805i \(-0.122634\pi\)
0.926699 + 0.375805i \(0.122634\pi\)
\(62\) 5.98594 0.760215
\(63\) 0 0
\(64\) 1.00000 0.125000
\(65\) 3.22455 0.399957
\(66\) 0 0
\(67\) 6.07141 0.741741 0.370870 0.928685i \(-0.379059\pi\)
0.370870 + 0.928685i \(0.379059\pi\)
\(68\) −1.35277 −0.164048
\(69\) 0 0
\(70\) 0.647226 0.0773583
\(71\) −8.22832 −0.976522 −0.488261 0.872698i \(-0.662369\pi\)
−0.488261 + 0.872698i \(0.662369\pi\)
\(72\) 0 0
\(73\) 8.08702 0.946514 0.473257 0.880924i \(-0.343078\pi\)
0.473257 + 0.880924i \(0.343078\pi\)
\(74\) −1.00000 −0.116248
\(75\) 0 0
\(76\) 6.30301 0.723005
\(77\) −2.38369 −0.271647
\(78\) 0 0
\(79\) 10.2283 1.15078 0.575388 0.817880i \(-0.304851\pi\)
0.575388 + 0.817880i \(0.304851\pi\)
\(80\) −1.00000 −0.111803
\(81\) 0 0
\(82\) −0.242382 −0.0267666
\(83\) 2.14131 0.235039 0.117520 0.993071i \(-0.462506\pi\)
0.117520 + 0.993071i \(0.462506\pi\)
\(84\) 0 0
\(85\) 1.35277 0.146729
\(86\) −1.75762 −0.189529
\(87\) 0 0
\(88\) 3.68293 0.392602
\(89\) 5.81346 0.616225 0.308113 0.951350i \(-0.400303\pi\)
0.308113 + 0.951350i \(0.400303\pi\)
\(90\) 0 0
\(91\) −2.08702 −0.218779
\(92\) −5.22455 −0.544697
\(93\) 0 0
\(94\) −6.44432 −0.664681
\(95\) −6.30301 −0.646675
\(96\) 0 0
\(97\) −4.20194 −0.426642 −0.213321 0.976982i \(-0.568428\pi\)
−0.213321 + 0.976982i \(0.568428\pi\)
\(98\) 6.58110 0.664791
\(99\) 0 0
\(100\) 1.00000 0.100000
\(101\) 17.3782 1.72919 0.864597 0.502465i \(-0.167574\pi\)
0.864597 + 0.502465i \(0.167574\pi\)
\(102\) 0 0
\(103\) 8.44432 0.832044 0.416022 0.909355i \(-0.363424\pi\)
0.416022 + 0.909355i \(0.363424\pi\)
\(104\) 3.22455 0.316194
\(105\) 0 0
\(106\) 5.46694 0.530996
\(107\) −12.9681 −1.25367 −0.626837 0.779150i \(-0.715651\pi\)
−0.626837 + 0.779150i \(0.715651\pi\)
\(108\) 0 0
\(109\) −9.91126 −0.949326 −0.474663 0.880168i \(-0.657430\pi\)
−0.474663 + 0.880168i \(0.657430\pi\)
\(110\) −3.68293 −0.351154
\(111\) 0 0
\(112\) 0.647226 0.0611571
\(113\) −11.2376 −1.05715 −0.528574 0.848887i \(-0.677273\pi\)
−0.528574 + 0.848887i \(0.677273\pi\)
\(114\) 0 0
\(115\) 5.22455 0.487192
\(116\) 3.87178 0.359486
\(117\) 0 0
\(118\) −7.36586 −0.678083
\(119\) −0.875551 −0.0802616
\(120\) 0 0
\(121\) 2.56398 0.233089
\(122\) −14.4755 −1.31055
\(123\) 0 0
\(124\) −5.98594 −0.537553
\(125\) −1.00000 −0.0894427
\(126\) 0 0
\(127\) 19.9020 1.76602 0.883008 0.469358i \(-0.155515\pi\)
0.883008 + 0.469358i \(0.155515\pi\)
\(128\) −1.00000 −0.0883883
\(129\) 0 0
\(130\) −3.22455 −0.282812
\(131\) −1.95477 −0.170789 −0.0853944 0.996347i \(-0.527215\pi\)
−0.0853944 + 0.996347i \(0.527215\pi\)
\(132\) 0 0
\(133\) 4.07948 0.353735
\(134\) −6.07141 −0.524490
\(135\) 0 0
\(136\) 1.35277 0.115999
\(137\) −1.78400 −0.152418 −0.0762089 0.997092i \(-0.524282\pi\)
−0.0762089 + 0.997092i \(0.524282\pi\)
\(138\) 0 0
\(139\) −1.00932 −0.0856092 −0.0428046 0.999083i \(-0.513629\pi\)
−0.0428046 + 0.999083i \(0.513629\pi\)
\(140\) −0.647226 −0.0547006
\(141\) 0 0
\(142\) 8.22832 0.690506
\(143\) 11.8758 0.993105
\(144\) 0 0
\(145\) −3.87178 −0.321534
\(146\) −8.08702 −0.669286
\(147\) 0 0
\(148\) 1.00000 0.0821995
\(149\) 20.4614 1.67627 0.838133 0.545466i \(-0.183647\pi\)
0.838133 + 0.545466i \(0.183647\pi\)
\(150\) 0 0
\(151\) −13.1964 −1.07391 −0.536955 0.843611i \(-0.680426\pi\)
−0.536955 + 0.843611i \(0.680426\pi\)
\(152\) −6.30301 −0.511242
\(153\) 0 0
\(154\) 2.38369 0.192083
\(155\) 5.98594 0.480802
\(156\) 0 0
\(157\) −8.24948 −0.658380 −0.329190 0.944264i \(-0.606776\pi\)
−0.329190 + 0.944264i \(0.606776\pi\)
\(158\) −10.2283 −0.813722
\(159\) 0 0
\(160\) 1.00000 0.0790569
\(161\) −3.38147 −0.266497
\(162\) 0 0
\(163\) −14.7785 −1.15754 −0.578771 0.815490i \(-0.696467\pi\)
−0.578771 + 0.815490i \(0.696467\pi\)
\(164\) 0.242382 0.0189269
\(165\) 0 0
\(166\) −2.14131 −0.166198
\(167\) 5.22455 0.404288 0.202144 0.979356i \(-0.435209\pi\)
0.202144 + 0.979356i \(0.435209\pi\)
\(168\) 0 0
\(169\) −2.60225 −0.200173
\(170\) −1.35277 −0.103753
\(171\) 0 0
\(172\) 1.75762 0.134017
\(173\) 20.9541 1.59311 0.796554 0.604568i \(-0.206654\pi\)
0.796554 + 0.604568i \(0.206654\pi\)
\(174\) 0 0
\(175\) 0.647226 0.0489257
\(176\) −3.68293 −0.277611
\(177\) 0 0
\(178\) −5.81346 −0.435737
\(179\) 9.77922 0.730933 0.365466 0.930824i \(-0.380909\pi\)
0.365466 + 0.930824i \(0.380909\pi\)
\(180\) 0 0
\(181\) 3.92859 0.292010 0.146005 0.989284i \(-0.453358\pi\)
0.146005 + 0.989284i \(0.453358\pi\)
\(182\) 2.08702 0.154700
\(183\) 0 0
\(184\) 5.22455 0.385159
\(185\) −1.00000 −0.0735215
\(186\) 0 0
\(187\) 4.98217 0.364332
\(188\) 6.44432 0.470000
\(189\) 0 0
\(190\) 6.30301 0.457269
\(191\) 6.86801 0.496952 0.248476 0.968638i \(-0.420070\pi\)
0.248476 + 0.968638i \(0.420070\pi\)
\(192\) 0 0
\(193\) 23.5537 1.69544 0.847718 0.530448i \(-0.177976\pi\)
0.847718 + 0.530448i \(0.177976\pi\)
\(194\) 4.20194 0.301682
\(195\) 0 0
\(196\) −6.58110 −0.470078
\(197\) −15.2960 −1.08979 −0.544896 0.838504i \(-0.683431\pi\)
−0.544896 + 0.838504i \(0.683431\pi\)
\(198\) 0 0
\(199\) −3.78879 −0.268580 −0.134290 0.990942i \(-0.542875\pi\)
−0.134290 + 0.990942i \(0.542875\pi\)
\(200\) −1.00000 −0.0707107
\(201\) 0 0
\(202\) −17.3782 −1.22273
\(203\) 2.50592 0.175881
\(204\) 0 0
\(205\) −0.242382 −0.0169287
\(206\) −8.44432 −0.588344
\(207\) 0 0
\(208\) −3.22455 −0.223583
\(209\) −23.2136 −1.60572
\(210\) 0 0
\(211\) 14.4778 0.996693 0.498347 0.866978i \(-0.333941\pi\)
0.498347 + 0.866978i \(0.333941\pi\)
\(212\) −5.46694 −0.375471
\(213\) 0 0
\(214\) 12.9681 0.886482
\(215\) −1.75762 −0.119869
\(216\) 0 0
\(217\) −3.87426 −0.263002
\(218\) 9.91126 0.671275
\(219\) 0 0
\(220\) 3.68293 0.248303
\(221\) 4.36209 0.293426
\(222\) 0 0
\(223\) 9.06209 0.606843 0.303421 0.952857i \(-0.401871\pi\)
0.303421 + 0.952857i \(0.401871\pi\)
\(224\) −0.647226 −0.0432446
\(225\) 0 0
\(226\) 11.2376 0.747517
\(227\) −7.94319 −0.527208 −0.263604 0.964631i \(-0.584911\pi\)
−0.263604 + 0.964631i \(0.584911\pi\)
\(228\) 0 0
\(229\) 23.0099 1.52054 0.760268 0.649609i \(-0.225067\pi\)
0.760268 + 0.649609i \(0.225067\pi\)
\(230\) −5.22455 −0.344497
\(231\) 0 0
\(232\) −3.87178 −0.254195
\(233\) −11.3707 −0.744916 −0.372458 0.928049i \(-0.621485\pi\)
−0.372458 + 0.928049i \(0.621485\pi\)
\(234\) 0 0
\(235\) −6.44432 −0.420381
\(236\) 7.36586 0.479477
\(237\) 0 0
\(238\) 0.875551 0.0567535
\(239\) 16.1453 1.04436 0.522178 0.852837i \(-0.325120\pi\)
0.522178 + 0.852837i \(0.325120\pi\)
\(240\) 0 0
\(241\) 29.4871 1.89943 0.949716 0.313112i \(-0.101372\pi\)
0.949716 + 0.313112i \(0.101372\pi\)
\(242\) −2.56398 −0.164819
\(243\) 0 0
\(244\) 14.4755 0.926699
\(245\) 6.58110 0.420451
\(246\) 0 0
\(247\) −20.3244 −1.29321
\(248\) 5.98594 0.380108
\(249\) 0 0
\(250\) 1.00000 0.0632456
\(251\) 7.85063 0.495527 0.247764 0.968821i \(-0.420304\pi\)
0.247764 + 0.968821i \(0.420304\pi\)
\(252\) 0 0
\(253\) 19.2417 1.20971
\(254\) −19.9020 −1.24876
\(255\) 0 0
\(256\) 1.00000 0.0625000
\(257\) 22.1846 1.38384 0.691919 0.721975i \(-0.256766\pi\)
0.691919 + 0.721975i \(0.256766\pi\)
\(258\) 0 0
\(259\) 0.647226 0.0402167
\(260\) 3.22455 0.199978
\(261\) 0 0
\(262\) 1.95477 0.120766
\(263\) 17.3395 1.06920 0.534599 0.845106i \(-0.320463\pi\)
0.534599 + 0.845106i \(0.320463\pi\)
\(264\) 0 0
\(265\) 5.46694 0.335831
\(266\) −4.07948 −0.250129
\(267\) 0 0
\(268\) 6.07141 0.370870
\(269\) −12.8235 −0.781864 −0.390932 0.920419i \(-0.627847\pi\)
−0.390932 + 0.920419i \(0.627847\pi\)
\(270\) 0 0
\(271\) 20.8886 1.26889 0.634447 0.772966i \(-0.281228\pi\)
0.634447 + 0.772966i \(0.281228\pi\)
\(272\) −1.35277 −0.0820239
\(273\) 0 0
\(274\) 1.78400 0.107776
\(275\) −3.68293 −0.222089
\(276\) 0 0
\(277\) −5.17932 −0.311195 −0.155598 0.987821i \(-0.549730\pi\)
−0.155598 + 0.987821i \(0.549730\pi\)
\(278\) 1.00932 0.0605349
\(279\) 0 0
\(280\) 0.647226 0.0386792
\(281\) 2.99623 0.178740 0.0893700 0.995998i \(-0.471515\pi\)
0.0893700 + 0.995998i \(0.471515\pi\)
\(282\) 0 0
\(283\) −11.8306 −0.703255 −0.351627 0.936140i \(-0.614372\pi\)
−0.351627 + 0.936140i \(0.614372\pi\)
\(284\) −8.22832 −0.488261
\(285\) 0 0
\(286\) −11.8758 −0.702231
\(287\) 0.156876 0.00926010
\(288\) 0 0
\(289\) −15.1700 −0.892353
\(290\) 3.87178 0.227359
\(291\) 0 0
\(292\) 8.08702 0.473257
\(293\) −8.55018 −0.499507 −0.249753 0.968309i \(-0.580350\pi\)
−0.249753 + 0.968309i \(0.580350\pi\)
\(294\) 0 0
\(295\) −7.36586 −0.428857
\(296\) −1.00000 −0.0581238
\(297\) 0 0
\(298\) −20.4614 −1.18530
\(299\) 16.8469 0.974279
\(300\) 0 0
\(301\) 1.13758 0.0655688
\(302\) 13.1964 0.759370
\(303\) 0 0
\(304\) 6.30301 0.361502
\(305\) −14.4755 −0.828864
\(306\) 0 0
\(307\) −11.3659 −0.648684 −0.324342 0.945940i \(-0.605143\pi\)
−0.324342 + 0.945940i \(0.605143\pi\)
\(308\) −2.38369 −0.135823
\(309\) 0 0
\(310\) −5.98594 −0.339979
\(311\) 17.3231 0.982306 0.491153 0.871073i \(-0.336576\pi\)
0.491153 + 0.871073i \(0.336576\pi\)
\(312\) 0 0
\(313\) 13.4777 0.761806 0.380903 0.924615i \(-0.375613\pi\)
0.380903 + 0.924615i \(0.375613\pi\)
\(314\) 8.24948 0.465545
\(315\) 0 0
\(316\) 10.2283 0.575388
\(317\) 4.96505 0.278865 0.139432 0.990232i \(-0.455472\pi\)
0.139432 + 0.990232i \(0.455472\pi\)
\(318\) 0 0
\(319\) −14.2595 −0.798379
\(320\) −1.00000 −0.0559017
\(321\) 0 0
\(322\) 3.38147 0.188442
\(323\) −8.52655 −0.474430
\(324\) 0 0
\(325\) −3.22455 −0.178866
\(326\) 14.7785 0.818506
\(327\) 0 0
\(328\) −0.242382 −0.0133833
\(329\) 4.17093 0.229951
\(330\) 0 0
\(331\) 6.97432 0.383343 0.191672 0.981459i \(-0.438609\pi\)
0.191672 + 0.981459i \(0.438609\pi\)
\(332\) 2.14131 0.117520
\(333\) 0 0
\(334\) −5.22455 −0.285875
\(335\) −6.07141 −0.331717
\(336\) 0 0
\(337\) −2.46916 −0.134504 −0.0672518 0.997736i \(-0.521423\pi\)
−0.0672518 + 0.997736i \(0.521423\pi\)
\(338\) 2.60225 0.141544
\(339\) 0 0
\(340\) 1.35277 0.0733645
\(341\) 22.0458 1.19385
\(342\) 0 0
\(343\) −8.79004 −0.474618
\(344\) −1.75762 −0.0947644
\(345\) 0 0
\(346\) −20.9541 −1.12650
\(347\) 5.03801 0.270455 0.135227 0.990815i \(-0.456824\pi\)
0.135227 + 0.990815i \(0.456824\pi\)
\(348\) 0 0
\(349\) −8.64168 −0.462578 −0.231289 0.972885i \(-0.574294\pi\)
−0.231289 + 0.972885i \(0.574294\pi\)
\(350\) −0.647226 −0.0345957
\(351\) 0 0
\(352\) 3.68293 0.196301
\(353\) −6.98120 −0.371572 −0.185786 0.982590i \(-0.559483\pi\)
−0.185786 + 0.982590i \(0.559483\pi\)
\(354\) 0 0
\(355\) 8.22832 0.436714
\(356\) 5.81346 0.308113
\(357\) 0 0
\(358\) −9.77922 −0.516848
\(359\) 34.4753 1.81954 0.909768 0.415117i \(-0.136259\pi\)
0.909768 + 0.415117i \(0.136259\pi\)
\(360\) 0 0
\(361\) 20.7280 1.09094
\(362\) −3.92859 −0.206482
\(363\) 0 0
\(364\) −2.08702 −0.109389
\(365\) −8.08702 −0.423294
\(366\) 0 0
\(367\) −26.1515 −1.36510 −0.682548 0.730841i \(-0.739128\pi\)
−0.682548 + 0.730841i \(0.739128\pi\)
\(368\) −5.22455 −0.272349
\(369\) 0 0
\(370\) 1.00000 0.0519875
\(371\) −3.53835 −0.183702
\(372\) 0 0
\(373\) −26.7488 −1.38500 −0.692501 0.721417i \(-0.743491\pi\)
−0.692501 + 0.721417i \(0.743491\pi\)
\(374\) −4.98217 −0.257622
\(375\) 0 0
\(376\) −6.44432 −0.332340
\(377\) −12.4848 −0.642998
\(378\) 0 0
\(379\) 6.74130 0.346277 0.173139 0.984897i \(-0.444609\pi\)
0.173139 + 0.984897i \(0.444609\pi\)
\(380\) −6.30301 −0.323338
\(381\) 0 0
\(382\) −6.86801 −0.351398
\(383\) 28.9354 1.47853 0.739264 0.673415i \(-0.235173\pi\)
0.739264 + 0.673415i \(0.235173\pi\)
\(384\) 0 0
\(385\) 2.38369 0.121484
\(386\) −23.5537 −1.19885
\(387\) 0 0
\(388\) −4.20194 −0.213321
\(389\) 28.0458 1.42198 0.710990 0.703203i \(-0.248247\pi\)
0.710990 + 0.703203i \(0.248247\pi\)
\(390\) 0 0
\(391\) 7.06764 0.357426
\(392\) 6.58110 0.332396
\(393\) 0 0
\(394\) 15.2960 0.770599
\(395\) −10.2283 −0.514643
\(396\) 0 0
\(397\) −29.0551 −1.45824 −0.729118 0.684388i \(-0.760069\pi\)
−0.729118 + 0.684388i \(0.760069\pi\)
\(398\) 3.78879 0.189915
\(399\) 0 0
\(400\) 1.00000 0.0500000
\(401\) −11.3131 −0.564948 −0.282474 0.959275i \(-0.591155\pi\)
−0.282474 + 0.959275i \(0.591155\pi\)
\(402\) 0 0
\(403\) 19.3020 0.961501
\(404\) 17.3782 0.864597
\(405\) 0 0
\(406\) −2.50592 −0.124367
\(407\) −3.68293 −0.182556
\(408\) 0 0
\(409\) 8.29973 0.410395 0.205198 0.978721i \(-0.434216\pi\)
0.205198 + 0.978721i \(0.434216\pi\)
\(410\) 0.242382 0.0119704
\(411\) 0 0
\(412\) 8.44432 0.416022
\(413\) 4.76738 0.234587
\(414\) 0 0
\(415\) −2.14131 −0.105113
\(416\) 3.22455 0.158097
\(417\) 0 0
\(418\) 23.2136 1.13541
\(419\) −13.3083 −0.650153 −0.325076 0.945688i \(-0.605390\pi\)
−0.325076 + 0.945688i \(0.605390\pi\)
\(420\) 0 0
\(421\) 15.7840 0.769265 0.384633 0.923070i \(-0.374328\pi\)
0.384633 + 0.923070i \(0.374328\pi\)
\(422\) −14.4778 −0.704768
\(423\) 0 0
\(424\) 5.46694 0.265498
\(425\) −1.35277 −0.0656192
\(426\) 0 0
\(427\) 9.36892 0.453394
\(428\) −12.9681 −0.626837
\(429\) 0 0
\(430\) 1.75762 0.0847599
\(431\) 27.7551 1.33692 0.668459 0.743749i \(-0.266954\pi\)
0.668459 + 0.743749i \(0.266954\pi\)
\(432\) 0 0
\(433\) −12.5190 −0.601625 −0.300813 0.953683i \(-0.597258\pi\)
−0.300813 + 0.953683i \(0.597258\pi\)
\(434\) 3.87426 0.185970
\(435\) 0 0
\(436\) −9.91126 −0.474663
\(437\) −32.9304 −1.57528
\(438\) 0 0
\(439\) −33.4946 −1.59861 −0.799306 0.600925i \(-0.794799\pi\)
−0.799306 + 0.600925i \(0.794799\pi\)
\(440\) −3.68293 −0.175577
\(441\) 0 0
\(442\) −4.36209 −0.207484
\(443\) 20.7131 0.984109 0.492054 0.870565i \(-0.336246\pi\)
0.492054 + 0.870565i \(0.336246\pi\)
\(444\) 0 0
\(445\) −5.81346 −0.275584
\(446\) −9.06209 −0.429102
\(447\) 0 0
\(448\) 0.647226 0.0305786
\(449\) 9.08325 0.428665 0.214332 0.976761i \(-0.431242\pi\)
0.214332 + 0.976761i \(0.431242\pi\)
\(450\) 0 0
\(451\) −0.892676 −0.0420345
\(452\) −11.2376 −0.528574
\(453\) 0 0
\(454\) 7.94319 0.372792
\(455\) 2.08702 0.0978408
\(456\) 0 0
\(457\) −10.8532 −0.507691 −0.253845 0.967245i \(-0.581695\pi\)
−0.253845 + 0.967245i \(0.581695\pi\)
\(458\) −23.0099 −1.07518
\(459\) 0 0
\(460\) 5.22455 0.243596
\(461\) −34.3455 −1.59963 −0.799816 0.600246i \(-0.795070\pi\)
−0.799816 + 0.600246i \(0.795070\pi\)
\(462\) 0 0
\(463\) 18.4087 0.855523 0.427762 0.903892i \(-0.359302\pi\)
0.427762 + 0.903892i \(0.359302\pi\)
\(464\) 3.87178 0.179743
\(465\) 0 0
\(466\) 11.3707 0.526735
\(467\) −28.2570 −1.30758 −0.653789 0.756677i \(-0.726822\pi\)
−0.653789 + 0.756677i \(0.726822\pi\)
\(468\) 0 0
\(469\) 3.92958 0.181451
\(470\) 6.44432 0.297254
\(471\) 0 0
\(472\) −7.36586 −0.339041
\(473\) −6.47319 −0.297637
\(474\) 0 0
\(475\) 6.30301 0.289202
\(476\) −0.875551 −0.0401308
\(477\) 0 0
\(478\) −16.1453 −0.738471
\(479\) 24.5642 1.12237 0.561184 0.827691i \(-0.310346\pi\)
0.561184 + 0.827691i \(0.310346\pi\)
\(480\) 0 0
\(481\) −3.22455 −0.147027
\(482\) −29.4871 −1.34310
\(483\) 0 0
\(484\) 2.56398 0.116545
\(485\) 4.20194 0.190800
\(486\) 0 0
\(487\) 9.22128 0.417856 0.208928 0.977931i \(-0.433003\pi\)
0.208928 + 0.977931i \(0.433003\pi\)
\(488\) −14.4755 −0.655275
\(489\) 0 0
\(490\) −6.58110 −0.297304
\(491\) 27.2415 1.22939 0.614696 0.788764i \(-0.289279\pi\)
0.614696 + 0.788764i \(0.289279\pi\)
\(492\) 0 0
\(493\) −5.23764 −0.235892
\(494\) 20.3244 0.914438
\(495\) 0 0
\(496\) −5.98594 −0.268777
\(497\) −5.32559 −0.238885
\(498\) 0 0
\(499\) 15.3083 0.685293 0.342647 0.939464i \(-0.388677\pi\)
0.342647 + 0.939464i \(0.388677\pi\)
\(500\) −1.00000 −0.0447214
\(501\) 0 0
\(502\) −7.85063 −0.350391
\(503\) 39.1979 1.74775 0.873875 0.486151i \(-0.161599\pi\)
0.873875 + 0.486151i \(0.161599\pi\)
\(504\) 0 0
\(505\) −17.3782 −0.773319
\(506\) −19.2417 −0.855397
\(507\) 0 0
\(508\) 19.9020 0.883008
\(509\) −23.6951 −1.05026 −0.525132 0.851021i \(-0.675984\pi\)
−0.525132 + 0.851021i \(0.675984\pi\)
\(510\) 0 0
\(511\) 5.23413 0.231544
\(512\) −1.00000 −0.0441942
\(513\) 0 0
\(514\) −22.1846 −0.978521
\(515\) −8.44432 −0.372101
\(516\) 0 0
\(517\) −23.7340 −1.04382
\(518\) −0.647226 −0.0284375
\(519\) 0 0
\(520\) −3.22455 −0.141406
\(521\) 39.0672 1.71157 0.855784 0.517334i \(-0.173075\pi\)
0.855784 + 0.517334i \(0.173075\pi\)
\(522\) 0 0
\(523\) −30.9867 −1.35496 −0.677478 0.735543i \(-0.736927\pi\)
−0.677478 + 0.735543i \(0.736927\pi\)
\(524\) −1.95477 −0.0853944
\(525\) 0 0
\(526\) −17.3395 −0.756037
\(527\) 8.09763 0.352738
\(528\) 0 0
\(529\) 4.29596 0.186781
\(530\) −5.46694 −0.237469
\(531\) 0 0
\(532\) 4.07948 0.176868
\(533\) −0.781574 −0.0338537
\(534\) 0 0
\(535\) 12.9681 0.560660
\(536\) −6.07141 −0.262245
\(537\) 0 0
\(538\) 12.8235 0.552862
\(539\) 24.2377 1.04399
\(540\) 0 0
\(541\) 18.7069 0.804272 0.402136 0.915580i \(-0.368268\pi\)
0.402136 + 0.915580i \(0.368268\pi\)
\(542\) −20.8886 −0.897244
\(543\) 0 0
\(544\) 1.35277 0.0579997
\(545\) 9.91126 0.424552
\(546\) 0 0
\(547\) 3.79186 0.162128 0.0810641 0.996709i \(-0.474168\pi\)
0.0810641 + 0.996709i \(0.474168\pi\)
\(548\) −1.78400 −0.0762089
\(549\) 0 0
\(550\) 3.68293 0.157041
\(551\) 24.4039 1.03964
\(552\) 0 0
\(553\) 6.62004 0.281513
\(554\) 5.17932 0.220048
\(555\) 0 0
\(556\) −1.00932 −0.0428046
\(557\) −25.2588 −1.07025 −0.535125 0.844773i \(-0.679735\pi\)
−0.535125 + 0.844773i \(0.679735\pi\)
\(558\) 0 0
\(559\) −5.66753 −0.239711
\(560\) −0.647226 −0.0273503
\(561\) 0 0
\(562\) −2.99623 −0.126388
\(563\) 13.3774 0.563792 0.281896 0.959445i \(-0.409037\pi\)
0.281896 + 0.959445i \(0.409037\pi\)
\(564\) 0 0
\(565\) 11.2376 0.472771
\(566\) 11.8306 0.497276
\(567\) 0 0
\(568\) 8.22832 0.345253
\(569\) 26.1303 1.09544 0.547720 0.836662i \(-0.315496\pi\)
0.547720 + 0.836662i \(0.315496\pi\)
\(570\) 0 0
\(571\) −4.60048 −0.192524 −0.0962621 0.995356i \(-0.530689\pi\)
−0.0962621 + 0.995356i \(0.530689\pi\)
\(572\) 11.8758 0.496553
\(573\) 0 0
\(574\) −0.156876 −0.00654788
\(575\) −5.22455 −0.217879
\(576\) 0 0
\(577\) 21.7107 0.903826 0.451913 0.892062i \(-0.350742\pi\)
0.451913 + 0.892062i \(0.350742\pi\)
\(578\) 15.1700 0.630989
\(579\) 0 0
\(580\) −3.87178 −0.160767
\(581\) 1.38591 0.0574973
\(582\) 0 0
\(583\) 20.1343 0.833879
\(584\) −8.08702 −0.334643
\(585\) 0 0
\(586\) 8.55018 0.353205
\(587\) 42.5925 1.75798 0.878990 0.476839i \(-0.158218\pi\)
0.878990 + 0.476839i \(0.158218\pi\)
\(588\) 0 0
\(589\) −37.7295 −1.55462
\(590\) 7.36586 0.303248
\(591\) 0 0
\(592\) 1.00000 0.0410997
\(593\) −19.7915 −0.812741 −0.406371 0.913708i \(-0.633206\pi\)
−0.406371 + 0.913708i \(0.633206\pi\)
\(594\) 0 0
\(595\) 0.875551 0.0358941
\(596\) 20.4614 0.838133
\(597\) 0 0
\(598\) −16.8469 −0.688919
\(599\) 10.0714 0.411507 0.205753 0.978604i \(-0.434036\pi\)
0.205753 + 0.978604i \(0.434036\pi\)
\(600\) 0 0
\(601\) −11.1677 −0.455542 −0.227771 0.973715i \(-0.573144\pi\)
−0.227771 + 0.973715i \(0.573144\pi\)
\(602\) −1.13758 −0.0463642
\(603\) 0 0
\(604\) −13.1964 −0.536955
\(605\) −2.56398 −0.104241
\(606\) 0 0
\(607\) 10.8653 0.441009 0.220505 0.975386i \(-0.429230\pi\)
0.220505 + 0.975386i \(0.429230\pi\)
\(608\) −6.30301 −0.255621
\(609\) 0 0
\(610\) 14.4755 0.586096
\(611\) −20.7801 −0.840671
\(612\) 0 0
\(613\) 22.8057 0.921112 0.460556 0.887631i \(-0.347650\pi\)
0.460556 + 0.887631i \(0.347650\pi\)
\(614\) 11.3659 0.458689
\(615\) 0 0
\(616\) 2.38369 0.0960416
\(617\) −23.6860 −0.953562 −0.476781 0.879022i \(-0.658197\pi\)
−0.476781 + 0.879022i \(0.658197\pi\)
\(618\) 0 0
\(619\) −46.7962 −1.88090 −0.940448 0.339936i \(-0.889595\pi\)
−0.940448 + 0.339936i \(0.889595\pi\)
\(620\) 5.98594 0.240401
\(621\) 0 0
\(622\) −17.3231 −0.694595
\(623\) 3.76262 0.150746
\(624\) 0 0
\(625\) 1.00000 0.0400000
\(626\) −13.4777 −0.538678
\(627\) 0 0
\(628\) −8.24948 −0.329190
\(629\) −1.35277 −0.0539386
\(630\) 0 0
\(631\) −28.1459 −1.12047 −0.560235 0.828334i \(-0.689289\pi\)
−0.560235 + 0.828334i \(0.689289\pi\)
\(632\) −10.2283 −0.406861
\(633\) 0 0
\(634\) −4.96505 −0.197187
\(635\) −19.9020 −0.789786
\(636\) 0 0
\(637\) 21.2211 0.840811
\(638\) 14.2595 0.564539
\(639\) 0 0
\(640\) 1.00000 0.0395285
\(641\) 1.48254 0.0585569 0.0292785 0.999571i \(-0.490679\pi\)
0.0292785 + 0.999571i \(0.490679\pi\)
\(642\) 0 0
\(643\) −35.3845 −1.39543 −0.697715 0.716376i \(-0.745800\pi\)
−0.697715 + 0.716376i \(0.745800\pi\)
\(644\) −3.38147 −0.133249
\(645\) 0 0
\(646\) 8.52655 0.335473
\(647\) −8.03717 −0.315974 −0.157987 0.987441i \(-0.550500\pi\)
−0.157987 + 0.987441i \(0.550500\pi\)
\(648\) 0 0
\(649\) −27.1280 −1.06487
\(650\) 3.22455 0.126477
\(651\) 0 0
\(652\) −14.7785 −0.578771
\(653\) 14.4135 0.564042 0.282021 0.959408i \(-0.408995\pi\)
0.282021 + 0.959408i \(0.408995\pi\)
\(654\) 0 0
\(655\) 1.95477 0.0763791
\(656\) 0.242382 0.00946343
\(657\) 0 0
\(658\) −4.17093 −0.162600
\(659\) −13.4995 −0.525864 −0.262932 0.964814i \(-0.584689\pi\)
−0.262932 + 0.964814i \(0.584689\pi\)
\(660\) 0 0
\(661\) −33.6005 −1.30691 −0.653454 0.756966i \(-0.726681\pi\)
−0.653454 + 0.756966i \(0.726681\pi\)
\(662\) −6.97432 −0.271064
\(663\) 0 0
\(664\) −2.14131 −0.0830989
\(665\) −4.07948 −0.158195
\(666\) 0 0
\(667\) −20.2283 −0.783244
\(668\) 5.22455 0.202144
\(669\) 0 0
\(670\) 6.07141 0.234559
\(671\) −53.3122 −2.05810
\(672\) 0 0
\(673\) 13.8396 0.533479 0.266739 0.963769i \(-0.414054\pi\)
0.266739 + 0.963769i \(0.414054\pi\)
\(674\) 2.46916 0.0951084
\(675\) 0 0
\(676\) −2.60225 −0.100087
\(677\) −42.1987 −1.62183 −0.810914 0.585166i \(-0.801030\pi\)
−0.810914 + 0.585166i \(0.801030\pi\)
\(678\) 0 0
\(679\) −2.71960 −0.104369
\(680\) −1.35277 −0.0518765
\(681\) 0 0
\(682\) −22.0458 −0.844178
\(683\) −35.4303 −1.35570 −0.677852 0.735199i \(-0.737089\pi\)
−0.677852 + 0.735199i \(0.737089\pi\)
\(684\) 0 0
\(685\) 1.78400 0.0681633
\(686\) 8.79004 0.335605
\(687\) 0 0
\(688\) 1.75762 0.0670086
\(689\) 17.6284 0.671590
\(690\) 0 0
\(691\) −10.8740 −0.413668 −0.206834 0.978376i \(-0.566316\pi\)
−0.206834 + 0.978376i \(0.566316\pi\)
\(692\) 20.9541 0.796554
\(693\) 0 0
\(694\) −5.03801 −0.191240
\(695\) 1.00932 0.0382856
\(696\) 0 0
\(697\) −0.327888 −0.0124196
\(698\) 8.64168 0.327092
\(699\) 0 0
\(700\) 0.647226 0.0244629
\(701\) −8.30629 −0.313724 −0.156862 0.987621i \(-0.550138\pi\)
−0.156862 + 0.987621i \(0.550138\pi\)
\(702\) 0 0
\(703\) 6.30301 0.237723
\(704\) −3.68293 −0.138806
\(705\) 0 0
\(706\) 6.98120 0.262741
\(707\) 11.2476 0.423010
\(708\) 0 0
\(709\) −10.5550 −0.396400 −0.198200 0.980162i \(-0.563510\pi\)
−0.198200 + 0.980162i \(0.563510\pi\)
\(710\) −8.22832 −0.308804
\(711\) 0 0
\(712\) −5.81346 −0.217869
\(713\) 31.2739 1.17122
\(714\) 0 0
\(715\) −11.8758 −0.444130
\(716\) 9.77922 0.365466
\(717\) 0 0
\(718\) −34.4753 −1.28661
\(719\) 24.4400 0.911460 0.455730 0.890118i \(-0.349378\pi\)
0.455730 + 0.890118i \(0.349378\pi\)
\(720\) 0 0
\(721\) 5.46539 0.203542
\(722\) −20.7280 −0.771415
\(723\) 0 0
\(724\) 3.92859 0.146005
\(725\) 3.87178 0.143794
\(726\) 0 0
\(727\) 12.8748 0.477500 0.238750 0.971081i \(-0.423262\pi\)
0.238750 + 0.971081i \(0.423262\pi\)
\(728\) 2.08702 0.0773500
\(729\) 0 0
\(730\) 8.08702 0.299314
\(731\) −2.37766 −0.0879409
\(732\) 0 0
\(733\) −26.5230 −0.979651 −0.489825 0.871821i \(-0.662939\pi\)
−0.489825 + 0.871821i \(0.662939\pi\)
\(734\) 26.1515 0.965268
\(735\) 0 0
\(736\) 5.22455 0.192580
\(737\) −22.3606 −0.823663
\(738\) 0 0
\(739\) −20.8599 −0.767345 −0.383673 0.923469i \(-0.625341\pi\)
−0.383673 + 0.923469i \(0.625341\pi\)
\(740\) −1.00000 −0.0367607
\(741\) 0 0
\(742\) 3.53835 0.129897
\(743\) −12.8794 −0.472498 −0.236249 0.971693i \(-0.575918\pi\)
−0.236249 + 0.971693i \(0.575918\pi\)
\(744\) 0 0
\(745\) −20.4614 −0.749649
\(746\) 26.7488 0.979345
\(747\) 0 0
\(748\) 4.98217 0.182166
\(749\) −8.39331 −0.306685
\(750\) 0 0
\(751\) −4.20367 −0.153394 −0.0766970 0.997054i \(-0.524437\pi\)
−0.0766970 + 0.997054i \(0.524437\pi\)
\(752\) 6.44432 0.235000
\(753\) 0 0
\(754\) 12.4848 0.454668
\(755\) 13.1964 0.480268
\(756\) 0 0
\(757\) −8.46623 −0.307710 −0.153855 0.988093i \(-0.549169\pi\)
−0.153855 + 0.988093i \(0.549169\pi\)
\(758\) −6.74130 −0.244855
\(759\) 0 0
\(760\) 6.30301 0.228634
\(761\) −10.3993 −0.376974 −0.188487 0.982076i \(-0.560358\pi\)
−0.188487 + 0.982076i \(0.560358\pi\)
\(762\) 0 0
\(763\) −6.41483 −0.232232
\(764\) 6.86801 0.248476
\(765\) 0 0
\(766\) −28.9354 −1.04548
\(767\) −23.7516 −0.857621
\(768\) 0 0
\(769\) 3.18083 0.114704 0.0573519 0.998354i \(-0.481734\pi\)
0.0573519 + 0.998354i \(0.481734\pi\)
\(770\) −2.38369 −0.0859022
\(771\) 0 0
\(772\) 23.5537 0.847718
\(773\) 35.7481 1.28577 0.642885 0.765962i \(-0.277737\pi\)
0.642885 + 0.765962i \(0.277737\pi\)
\(774\) 0 0
\(775\) −5.98594 −0.215021
\(776\) 4.20194 0.150841
\(777\) 0 0
\(778\) −28.0458 −1.00549
\(779\) 1.52774 0.0547368
\(780\) 0 0
\(781\) 30.3044 1.08437
\(782\) −7.06764 −0.252738
\(783\) 0 0
\(784\) −6.58110 −0.235039
\(785\) 8.24948 0.294437
\(786\) 0 0
\(787\) −24.1474 −0.860763 −0.430382 0.902647i \(-0.641621\pi\)
−0.430382 + 0.902647i \(0.641621\pi\)
\(788\) −15.2960 −0.544896
\(789\) 0 0
\(790\) 10.2283 0.363907
\(791\) −7.27330 −0.258609
\(792\) 0 0
\(793\) −46.6770 −1.65755
\(794\) 29.0551 1.03113
\(795\) 0 0
\(796\) −3.78879 −0.134290
\(797\) −28.4753 −1.00865 −0.504323 0.863515i \(-0.668258\pi\)
−0.504323 + 0.863515i \(0.668258\pi\)
\(798\) 0 0
\(799\) −8.71771 −0.308410
\(800\) −1.00000 −0.0353553
\(801\) 0 0
\(802\) 11.3131 0.399479
\(803\) −29.7839 −1.05105
\(804\) 0 0
\(805\) 3.38147 0.119181
\(806\) −19.3020 −0.679884
\(807\) 0 0
\(808\) −17.3782 −0.611363
\(809\) −25.9825 −0.913494 −0.456747 0.889597i \(-0.650986\pi\)
−0.456747 + 0.889597i \(0.650986\pi\)
\(810\) 0 0
\(811\) 28.2360 0.991499 0.495749 0.868466i \(-0.334893\pi\)
0.495749 + 0.868466i \(0.334893\pi\)
\(812\) 2.50592 0.0879405
\(813\) 0 0
\(814\) 3.68293 0.129087
\(815\) 14.7785 0.517668
\(816\) 0 0
\(817\) 11.0783 0.387580
\(818\) −8.29973 −0.290193
\(819\) 0 0
\(820\) −0.242382 −0.00846435
\(821\) −31.8087 −1.11013 −0.555065 0.831807i \(-0.687307\pi\)
−0.555065 + 0.831807i \(0.687307\pi\)
\(822\) 0 0
\(823\) −10.3169 −0.359623 −0.179812 0.983701i \(-0.557549\pi\)
−0.179812 + 0.983701i \(0.557549\pi\)
\(824\) −8.44432 −0.294172
\(825\) 0 0
\(826\) −4.76738 −0.165878
\(827\) −22.6487 −0.787574 −0.393787 0.919202i \(-0.628835\pi\)
−0.393787 + 0.919202i \(0.628835\pi\)
\(828\) 0 0
\(829\) −39.8071 −1.38256 −0.691279 0.722588i \(-0.742952\pi\)
−0.691279 + 0.722588i \(0.742952\pi\)
\(830\) 2.14131 0.0743259
\(831\) 0 0
\(832\) −3.22455 −0.111791
\(833\) 8.90274 0.308462
\(834\) 0 0
\(835\) −5.22455 −0.180803
\(836\) −23.2136 −0.802858
\(837\) 0 0
\(838\) 13.3083 0.459727
\(839\) −3.37544 −0.116533 −0.0582665 0.998301i \(-0.518557\pi\)
−0.0582665 + 0.998301i \(0.518557\pi\)
\(840\) 0 0
\(841\) −14.0093 −0.483080
\(842\) −15.7840 −0.543953
\(843\) 0 0
\(844\) 14.4778 0.498347
\(845\) 2.60225 0.0895202
\(846\) 0 0
\(847\) 1.65948 0.0570203
\(848\) −5.46694 −0.187735
\(849\) 0 0
\(850\) 1.35277 0.0463998
\(851\) −5.22455 −0.179095
\(852\) 0 0
\(853\) 33.2417 1.13817 0.569087 0.822278i \(-0.307297\pi\)
0.569087 + 0.822278i \(0.307297\pi\)
\(854\) −9.36892 −0.320598
\(855\) 0 0
\(856\) 12.9681 0.443241
\(857\) −23.0465 −0.787253 −0.393626 0.919270i \(-0.628780\pi\)
−0.393626 + 0.919270i \(0.628780\pi\)
\(858\) 0 0
\(859\) −40.6123 −1.38567 −0.692837 0.721094i \(-0.743640\pi\)
−0.692837 + 0.721094i \(0.743640\pi\)
\(860\) −1.75762 −0.0599343
\(861\) 0 0
\(862\) −27.7551 −0.945344
\(863\) 30.4021 1.03490 0.517451 0.855713i \(-0.326881\pi\)
0.517451 + 0.855713i \(0.326881\pi\)
\(864\) 0 0
\(865\) −20.9541 −0.712459
\(866\) 12.5190 0.425413
\(867\) 0 0
\(868\) −3.87426 −0.131501
\(869\) −37.6702 −1.27787
\(870\) 0 0
\(871\) −19.5776 −0.663361
\(872\) 9.91126 0.335637
\(873\) 0 0
\(874\) 32.9304 1.11389
\(875\) −0.647226 −0.0218802
\(876\) 0 0
\(877\) −48.9721 −1.65367 −0.826836 0.562443i \(-0.809861\pi\)
−0.826836 + 0.562443i \(0.809861\pi\)
\(878\) 33.4946 1.13039
\(879\) 0 0
\(880\) 3.68293 0.124152
\(881\) −9.31809 −0.313934 −0.156967 0.987604i \(-0.550172\pi\)
−0.156967 + 0.987604i \(0.550172\pi\)
\(882\) 0 0
\(883\) 19.5555 0.658094 0.329047 0.944314i \(-0.393273\pi\)
0.329047 + 0.944314i \(0.393273\pi\)
\(884\) 4.36209 0.146713
\(885\) 0 0
\(886\) −20.7131 −0.695870
\(887\) 9.95507 0.334259 0.167129 0.985935i \(-0.446550\pi\)
0.167129 + 0.985935i \(0.446550\pi\)
\(888\) 0 0
\(889\) 12.8811 0.432018
\(890\) 5.81346 0.194868
\(891\) 0 0
\(892\) 9.06209 0.303421
\(893\) 40.6186 1.35925
\(894\) 0 0
\(895\) −9.77922 −0.326883
\(896\) −0.647226 −0.0216223
\(897\) 0 0
\(898\) −9.08325 −0.303112
\(899\) −23.1763 −0.772971
\(900\) 0 0
\(901\) 7.39553 0.246381
\(902\) 0.892676 0.0297229
\(903\) 0 0
\(904\) 11.2376 0.373758
\(905\) −3.92859 −0.130591
\(906\) 0 0
\(907\) 50.2797 1.66951 0.834755 0.550622i \(-0.185609\pi\)
0.834755 + 0.550622i \(0.185609\pi\)
\(908\) −7.94319 −0.263604
\(909\) 0 0
\(910\) −2.08702 −0.0691839
\(911\) −23.0456 −0.763533 −0.381767 0.924259i \(-0.624684\pi\)
−0.381767 + 0.924259i \(0.624684\pi\)
\(912\) 0 0
\(913\) −7.88629 −0.260998
\(914\) 10.8532 0.358992
\(915\) 0 0
\(916\) 23.0099 0.760268
\(917\) −1.26518 −0.0417798
\(918\) 0 0
\(919\) −19.3252 −0.637481 −0.318741 0.947842i \(-0.603260\pi\)
−0.318741 + 0.947842i \(0.603260\pi\)
\(920\) −5.22455 −0.172248
\(921\) 0 0
\(922\) 34.3455 1.13111
\(923\) 26.5327 0.873334
\(924\) 0 0
\(925\) 1.00000 0.0328798
\(926\) −18.4087 −0.604946
\(927\) 0 0
\(928\) −3.87178 −0.127097
\(929\) 20.3013 0.666064 0.333032 0.942916i \(-0.391928\pi\)
0.333032 + 0.942916i \(0.391928\pi\)
\(930\) 0 0
\(931\) −41.4807 −1.35948
\(932\) −11.3707 −0.372458
\(933\) 0 0
\(934\) 28.2570 0.924598
\(935\) −4.98217 −0.162934
\(936\) 0 0
\(937\) 5.08786 0.166213 0.0831066 0.996541i \(-0.473516\pi\)
0.0831066 + 0.996541i \(0.473516\pi\)
\(938\) −3.92958 −0.128305
\(939\) 0 0
\(940\) −6.44432 −0.210191
\(941\) 28.8079 0.939111 0.469556 0.882903i \(-0.344414\pi\)
0.469556 + 0.882903i \(0.344414\pi\)
\(942\) 0 0
\(943\) −1.26634 −0.0412376
\(944\) 7.36586 0.239738
\(945\) 0 0
\(946\) 6.47319 0.210461
\(947\) −35.4303 −1.15133 −0.575665 0.817686i \(-0.695257\pi\)
−0.575665 + 0.817686i \(0.695257\pi\)
\(948\) 0 0
\(949\) −26.0770 −0.846496
\(950\) −6.30301 −0.204497
\(951\) 0 0
\(952\) 0.875551 0.0283768
\(953\) −14.8501 −0.481043 −0.240521 0.970644i \(-0.577318\pi\)
−0.240521 + 0.970644i \(0.577318\pi\)
\(954\) 0 0
\(955\) −6.86801 −0.222244
\(956\) 16.1453 0.522178
\(957\) 0 0
\(958\) −24.5642 −0.793635
\(959\) −1.15465 −0.0372858
\(960\) 0 0
\(961\) 4.83151 0.155855
\(962\) 3.22455 0.103964
\(963\) 0 0
\(964\) 29.4871 0.949716
\(965\) −23.5537 −0.758222
\(966\) 0 0
\(967\) 0.593694 0.0190919 0.00954595 0.999954i \(-0.496961\pi\)
0.00954595 + 0.999954i \(0.496961\pi\)
\(968\) −2.56398 −0.0824095
\(969\) 0 0
\(970\) −4.20194 −0.134916
\(971\) −31.8152 −1.02100 −0.510499 0.859878i \(-0.670539\pi\)
−0.510499 + 0.859878i \(0.670539\pi\)
\(972\) 0 0
\(973\) −0.653257 −0.0209425
\(974\) −9.22128 −0.295469
\(975\) 0 0
\(976\) 14.4755 0.463349
\(977\) 7.05606 0.225743 0.112872 0.993610i \(-0.463995\pi\)
0.112872 + 0.993610i \(0.463995\pi\)
\(978\) 0 0
\(979\) −21.4106 −0.684285
\(980\) 6.58110 0.210225
\(981\) 0 0
\(982\) −27.2415 −0.869311
\(983\) −4.73957 −0.151169 −0.0755844 0.997139i \(-0.524082\pi\)
−0.0755844 + 0.997139i \(0.524082\pi\)
\(984\) 0 0
\(985\) 15.2960 0.487370
\(986\) 5.23764 0.166801
\(987\) 0 0
\(988\) −20.3244 −0.646605
\(989\) −9.18277 −0.291995
\(990\) 0 0
\(991\) 30.2404 0.960619 0.480310 0.877099i \(-0.340524\pi\)
0.480310 + 0.877099i \(0.340524\pi\)
\(992\) 5.98594 0.190054
\(993\) 0 0
\(994\) 5.32559 0.168917
\(995\) 3.78879 0.120113
\(996\) 0 0
\(997\) 56.4939 1.78918 0.894590 0.446888i \(-0.147468\pi\)
0.894590 + 0.446888i \(0.147468\pi\)
\(998\) −15.3083 −0.484576
\(999\) 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 3330.2.a.bk.1.3 5
3.2 odd 2 3330.2.a.bl.1.3 yes 5
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
3330.2.a.bk.1.3 5 1.1 even 1 trivial
3330.2.a.bl.1.3 yes 5 3.2 odd 2