Properties

Label 2151.2.a.j.1.10
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $20$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 2151 = 3^{2} \cdot 239 \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 2151.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(17.1758214748\)
Analytic rank: \(1\)
Dimension: \(20\)
Coefficient field: \(\mathbb{Q}[x]/(x^{20} - \cdots)\)
Defining polynomial: \(x^{20} - 4 x^{19} - 21 x^{18} + 96 x^{17} + 164 x^{16} - 936 x^{15} - 540 x^{14} + 4804 x^{13} + 229 x^{12} - 14020 x^{11} + 3356 x^{10} + 23404 x^{9} - 9429 x^{8} - 21252 x^{7} + 10479 x^{6} + 9108 x^{5} - 4844 x^{4} - 1184 x^{3} + 640 x^{2} - 56 x + 1\)
Coefficient ring: \(\Z[a_1, \ldots, a_{7}]\)
Coefficient ring index: \( 2^{2} \)
Twist minimal: yes
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.10
Root \(0.244320\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.244320 q^{2} -1.94031 q^{4} -4.22460 q^{5} +0.0748971 q^{7} +0.962697 q^{8} +O(q^{10})\) \(q-0.244320 q^{2} -1.94031 q^{4} -4.22460 q^{5} +0.0748971 q^{7} +0.962697 q^{8} +1.03216 q^{10} -3.64400 q^{11} +6.36423 q^{13} -0.0182989 q^{14} +3.64541 q^{16} +2.35633 q^{17} +1.06774 q^{19} +8.19702 q^{20} +0.890304 q^{22} +1.33116 q^{23} +12.8473 q^{25} -1.55491 q^{26} -0.145323 q^{28} +8.25717 q^{29} -7.90166 q^{31} -2.81604 q^{32} -0.575699 q^{34} -0.316410 q^{35} -2.87745 q^{37} -0.260871 q^{38} -4.06701 q^{40} -7.12557 q^{41} +6.29063 q^{43} +7.07048 q^{44} -0.325230 q^{46} -6.74450 q^{47} -6.99439 q^{49} -3.13884 q^{50} -12.3486 q^{52} -9.74812 q^{53} +15.3945 q^{55} +0.0721032 q^{56} -2.01739 q^{58} +4.34109 q^{59} -13.4316 q^{61} +1.93054 q^{62} -6.60280 q^{64} -26.8863 q^{65} -4.26661 q^{67} -4.57200 q^{68} +0.0773054 q^{70} +14.0313 q^{71} +6.66218 q^{73} +0.703019 q^{74} -2.07174 q^{76} -0.272925 q^{77} +17.5165 q^{79} -15.4004 q^{80} +1.74092 q^{82} -11.0622 q^{83} -9.95454 q^{85} -1.53693 q^{86} -3.50807 q^{88} -10.5732 q^{89} +0.476662 q^{91} -2.58286 q^{92} +1.64782 q^{94} -4.51078 q^{95} +9.62047 q^{97} +1.70887 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + O(q^{10}) \) \( 20q - 4q^{2} + 18q^{4} - 16q^{5} - 4q^{7} - 12q^{8} + 4q^{10} - 12q^{11} - 4q^{13} - 20q^{14} + 22q^{16} - 24q^{17} - 4q^{19} - 40q^{20} - 6q^{22} - 12q^{23} + 22q^{25} - 30q^{26} - 12q^{28} - 24q^{29} - 4q^{31} - 28q^{32} + 8q^{34} - 20q^{35} - 10q^{37} - 26q^{38} + 6q^{40} - 66q^{41} + 8q^{43} - 36q^{44} - 12q^{46} - 28q^{47} + 18q^{49} - 28q^{50} - 18q^{52} - 28q^{53} - 4q^{55} - 60q^{56} - 54q^{59} - 4q^{61} - 20q^{62} + 22q^{64} - 42q^{65} + 12q^{67} - 12q^{68} + 20q^{70} - 36q^{71} + 14q^{73} - 50q^{76} - 8q^{77} - 12q^{79} - 88q^{80} - 8q^{82} - 20q^{83} + 4q^{85} - 18q^{86} - 10q^{88} - 130q^{89} - 6q^{91} + 46q^{92} - 26q^{94} - 2q^{97} - 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.244320 −0.172761 −0.0863803 0.996262i \(-0.527530\pi\)
−0.0863803 + 0.996262i \(0.527530\pi\)
\(3\) 0 0
\(4\) −1.94031 −0.970154
\(5\) −4.22460 −1.88930 −0.944649 0.328081i \(-0.893598\pi\)
−0.944649 + 0.328081i \(0.893598\pi\)
\(6\) 0 0
\(7\) 0.0748971 0.0283084 0.0141542 0.999900i \(-0.495494\pi\)
0.0141542 + 0.999900i \(0.495494\pi\)
\(8\) 0.962697 0.340365
\(9\) 0 0
\(10\) 1.03216 0.326396
\(11\) −3.64400 −1.09871 −0.549354 0.835590i \(-0.685126\pi\)
−0.549354 + 0.835590i \(0.685126\pi\)
\(12\) 0 0
\(13\) 6.36423 1.76512 0.882560 0.470200i \(-0.155818\pi\)
0.882560 + 0.470200i \(0.155818\pi\)
\(14\) −0.0182989 −0.00489058
\(15\) 0 0
\(16\) 3.64541 0.911352
\(17\) 2.35633 0.571493 0.285747 0.958305i \(-0.407758\pi\)
0.285747 + 0.958305i \(0.407758\pi\)
\(18\) 0 0
\(19\) 1.06774 0.244956 0.122478 0.992471i \(-0.460916\pi\)
0.122478 + 0.992471i \(0.460916\pi\)
\(20\) 8.19702 1.83291
\(21\) 0 0
\(22\) 0.890304 0.189813
\(23\) 1.33116 0.277566 0.138783 0.990323i \(-0.455681\pi\)
0.138783 + 0.990323i \(0.455681\pi\)
\(24\) 0 0
\(25\) 12.8473 2.56945
\(26\) −1.55491 −0.304943
\(27\) 0 0
\(28\) −0.145323 −0.0274635
\(29\) 8.25717 1.53332 0.766659 0.642054i \(-0.221918\pi\)
0.766659 + 0.642054i \(0.221918\pi\)
\(30\) 0 0
\(31\) −7.90166 −1.41918 −0.709590 0.704615i \(-0.751120\pi\)
−0.709590 + 0.704615i \(0.751120\pi\)
\(32\) −2.81604 −0.497811
\(33\) 0 0
\(34\) −0.575699 −0.0987315
\(35\) −0.316410 −0.0534831
\(36\) 0 0
\(37\) −2.87745 −0.473050 −0.236525 0.971625i \(-0.576008\pi\)
−0.236525 + 0.971625i \(0.576008\pi\)
\(38\) −0.260871 −0.0423188
\(39\) 0 0
\(40\) −4.06701 −0.643051
\(41\) −7.12557 −1.11283 −0.556414 0.830905i \(-0.687823\pi\)
−0.556414 + 0.830905i \(0.687823\pi\)
\(42\) 0 0
\(43\) 6.29063 0.959313 0.479656 0.877456i \(-0.340761\pi\)
0.479656 + 0.877456i \(0.340761\pi\)
\(44\) 7.07048 1.06592
\(45\) 0 0
\(46\) −0.325230 −0.0479525
\(47\) −6.74450 −0.983786 −0.491893 0.870656i \(-0.663695\pi\)
−0.491893 + 0.870656i \(0.663695\pi\)
\(48\) 0 0
\(49\) −6.99439 −0.999199
\(50\) −3.13884 −0.443900
\(51\) 0 0
\(52\) −12.3486 −1.71244
\(53\) −9.74812 −1.33901 −0.669504 0.742809i \(-0.733493\pi\)
−0.669504 + 0.742809i \(0.733493\pi\)
\(54\) 0 0
\(55\) 15.3945 2.07579
\(56\) 0.0721032 0.00963519
\(57\) 0 0
\(58\) −2.01739 −0.264897
\(59\) 4.34109 0.565162 0.282581 0.959243i \(-0.408809\pi\)
0.282581 + 0.959243i \(0.408809\pi\)
\(60\) 0 0
\(61\) −13.4316 −1.71974 −0.859870 0.510513i \(-0.829456\pi\)
−0.859870 + 0.510513i \(0.829456\pi\)
\(62\) 1.93054 0.245178
\(63\) 0 0
\(64\) −6.60280 −0.825350
\(65\) −26.8863 −3.33484
\(66\) 0 0
\(67\) −4.26661 −0.521249 −0.260625 0.965440i \(-0.583928\pi\)
−0.260625 + 0.965440i \(0.583928\pi\)
\(68\) −4.57200 −0.554436
\(69\) 0 0
\(70\) 0.0773054 0.00923977
\(71\) 14.0313 1.66521 0.832604 0.553869i \(-0.186849\pi\)
0.832604 + 0.553869i \(0.186849\pi\)
\(72\) 0 0
\(73\) 6.66218 0.779749 0.389874 0.920868i \(-0.372518\pi\)
0.389874 + 0.920868i \(0.372518\pi\)
\(74\) 0.703019 0.0817243
\(75\) 0 0
\(76\) −2.07174 −0.237645
\(77\) −0.272925 −0.0311027
\(78\) 0 0
\(79\) 17.5165 1.97077 0.985383 0.170356i \(-0.0544918\pi\)
0.985383 + 0.170356i \(0.0544918\pi\)
\(80\) −15.4004 −1.72182
\(81\) 0 0
\(82\) 1.74092 0.192253
\(83\) −11.0622 −1.21423 −0.607117 0.794613i \(-0.707674\pi\)
−0.607117 + 0.794613i \(0.707674\pi\)
\(84\) 0 0
\(85\) −9.95454 −1.07972
\(86\) −1.53693 −0.165731
\(87\) 0 0
\(88\) −3.50807 −0.373961
\(89\) −10.5732 −1.12075 −0.560376 0.828238i \(-0.689343\pi\)
−0.560376 + 0.828238i \(0.689343\pi\)
\(90\) 0 0
\(91\) 0.476662 0.0499678
\(92\) −2.58286 −0.269282
\(93\) 0 0
\(94\) 1.64782 0.169959
\(95\) −4.51078 −0.462796
\(96\) 0 0
\(97\) 9.62047 0.976810 0.488405 0.872617i \(-0.337579\pi\)
0.488405 + 0.872617i \(0.337579\pi\)
\(98\) 1.70887 0.172622
\(99\) 0 0
\(100\) −24.9276 −2.49276
\(101\) 8.02921 0.798936 0.399468 0.916747i \(-0.369195\pi\)
0.399468 + 0.916747i \(0.369195\pi\)
\(102\) 0 0
\(103\) 1.57808 0.155493 0.0777465 0.996973i \(-0.475228\pi\)
0.0777465 + 0.996973i \(0.475228\pi\)
\(104\) 6.12683 0.600785
\(105\) 0 0
\(106\) 2.38166 0.231328
\(107\) 17.8549 1.72609 0.863047 0.505123i \(-0.168553\pi\)
0.863047 + 0.505123i \(0.168553\pi\)
\(108\) 0 0
\(109\) 3.68362 0.352827 0.176414 0.984316i \(-0.443550\pi\)
0.176414 + 0.984316i \(0.443550\pi\)
\(110\) −3.76118 −0.358614
\(111\) 0 0
\(112\) 0.273030 0.0257989
\(113\) −17.8113 −1.67555 −0.837775 0.546016i \(-0.816144\pi\)
−0.837775 + 0.546016i \(0.816144\pi\)
\(114\) 0 0
\(115\) −5.62362 −0.524405
\(116\) −16.0215 −1.48755
\(117\) 0 0
\(118\) −1.06062 −0.0976377
\(119\) 0.176482 0.0161781
\(120\) 0 0
\(121\) 2.27875 0.207159
\(122\) 3.28161 0.297103
\(123\) 0 0
\(124\) 15.3316 1.37682
\(125\) −33.1515 −2.96516
\(126\) 0 0
\(127\) −10.4238 −0.924959 −0.462479 0.886630i \(-0.653040\pi\)
−0.462479 + 0.886630i \(0.653040\pi\)
\(128\) 7.24528 0.640398
\(129\) 0 0
\(130\) 6.56888 0.576129
\(131\) 1.77066 0.154703 0.0773514 0.997004i \(-0.475354\pi\)
0.0773514 + 0.997004i \(0.475354\pi\)
\(132\) 0 0
\(133\) 0.0799706 0.00693433
\(134\) 1.04242 0.0900513
\(135\) 0 0
\(136\) 2.26843 0.194516
\(137\) −6.86164 −0.586229 −0.293115 0.956077i \(-0.594692\pi\)
−0.293115 + 0.956077i \(0.594692\pi\)
\(138\) 0 0
\(139\) 8.85784 0.751312 0.375656 0.926759i \(-0.377417\pi\)
0.375656 + 0.926759i \(0.377417\pi\)
\(140\) 0.613933 0.0518868
\(141\) 0 0
\(142\) −3.42813 −0.287682
\(143\) −23.1913 −1.93935
\(144\) 0 0
\(145\) −34.8832 −2.89690
\(146\) −1.62770 −0.134710
\(147\) 0 0
\(148\) 5.58314 0.458931
\(149\) −8.32832 −0.682283 −0.341141 0.940012i \(-0.610813\pi\)
−0.341141 + 0.940012i \(0.610813\pi\)
\(150\) 0 0
\(151\) −23.9749 −1.95105 −0.975527 0.219881i \(-0.929433\pi\)
−0.975527 + 0.219881i \(0.929433\pi\)
\(152\) 1.02791 0.0833746
\(153\) 0 0
\(154\) 0.0666811 0.00537332
\(155\) 33.3813 2.68125
\(156\) 0 0
\(157\) −7.07989 −0.565037 −0.282518 0.959262i \(-0.591170\pi\)
−0.282518 + 0.959262i \(0.591170\pi\)
\(158\) −4.27965 −0.340470
\(159\) 0 0
\(160\) 11.8967 0.940513
\(161\) 0.0997000 0.00785746
\(162\) 0 0
\(163\) −5.72901 −0.448731 −0.224366 0.974505i \(-0.572031\pi\)
−0.224366 + 0.974505i \(0.572031\pi\)
\(164\) 13.8258 1.07961
\(165\) 0 0
\(166\) 2.70272 0.209772
\(167\) 11.6739 0.903356 0.451678 0.892181i \(-0.350826\pi\)
0.451678 + 0.892181i \(0.350826\pi\)
\(168\) 0 0
\(169\) 27.5034 2.11565
\(170\) 2.43210 0.186533
\(171\) 0 0
\(172\) −12.2058 −0.930681
\(173\) 4.00079 0.304174 0.152087 0.988367i \(-0.451401\pi\)
0.152087 + 0.988367i \(0.451401\pi\)
\(174\) 0 0
\(175\) 0.962221 0.0727371
\(176\) −13.2839 −1.00131
\(177\) 0 0
\(178\) 2.58324 0.193622
\(179\) −18.0881 −1.35197 −0.675985 0.736915i \(-0.736282\pi\)
−0.675985 + 0.736915i \(0.736282\pi\)
\(180\) 0 0
\(181\) 12.0863 0.898366 0.449183 0.893440i \(-0.351715\pi\)
0.449183 + 0.893440i \(0.351715\pi\)
\(182\) −0.116458 −0.00863246
\(183\) 0 0
\(184\) 1.28150 0.0944738
\(185\) 12.1561 0.893732
\(186\) 0 0
\(187\) −8.58646 −0.627904
\(188\) 13.0864 0.954424
\(189\) 0 0
\(190\) 1.10207 0.0799529
\(191\) 8.49649 0.614785 0.307392 0.951583i \(-0.400544\pi\)
0.307392 + 0.951583i \(0.400544\pi\)
\(192\) 0 0
\(193\) −14.8139 −1.06633 −0.533164 0.846012i \(-0.678997\pi\)
−0.533164 + 0.846012i \(0.678997\pi\)
\(194\) −2.35048 −0.168754
\(195\) 0 0
\(196\) 13.5713 0.969376
\(197\) −3.15281 −0.224629 −0.112314 0.993673i \(-0.535826\pi\)
−0.112314 + 0.993673i \(0.535826\pi\)
\(198\) 0 0
\(199\) −16.0694 −1.13913 −0.569563 0.821947i \(-0.692888\pi\)
−0.569563 + 0.821947i \(0.692888\pi\)
\(200\) 12.3680 0.874551
\(201\) 0 0
\(202\) −1.96170 −0.138025
\(203\) 0.618438 0.0434058
\(204\) 0 0
\(205\) 30.1027 2.10246
\(206\) −0.385557 −0.0268630
\(207\) 0 0
\(208\) 23.2002 1.60865
\(209\) −3.89085 −0.269136
\(210\) 0 0
\(211\) 16.8371 1.15912 0.579558 0.814931i \(-0.303225\pi\)
0.579558 + 0.814931i \(0.303225\pi\)
\(212\) 18.9144 1.29904
\(213\) 0 0
\(214\) −4.36231 −0.298201
\(215\) −26.5754 −1.81243
\(216\) 0 0
\(217\) −0.591811 −0.0401747
\(218\) −0.899984 −0.0609546
\(219\) 0 0
\(220\) −29.8700 −2.01383
\(221\) 14.9962 1.00875
\(222\) 0 0
\(223\) −21.5486 −1.44300 −0.721502 0.692413i \(-0.756548\pi\)
−0.721502 + 0.692413i \(0.756548\pi\)
\(224\) −0.210913 −0.0140922
\(225\) 0 0
\(226\) 4.35167 0.289469
\(227\) −17.1428 −1.13781 −0.568903 0.822405i \(-0.692632\pi\)
−0.568903 + 0.822405i \(0.692632\pi\)
\(228\) 0 0
\(229\) −13.0286 −0.860956 −0.430478 0.902601i \(-0.641655\pi\)
−0.430478 + 0.902601i \(0.641655\pi\)
\(230\) 1.37396 0.0905966
\(231\) 0 0
\(232\) 7.94915 0.521888
\(233\) −18.0528 −1.18268 −0.591338 0.806424i \(-0.701400\pi\)
−0.591338 + 0.806424i \(0.701400\pi\)
\(234\) 0 0
\(235\) 28.4928 1.85867
\(236\) −8.42305 −0.548294
\(237\) 0 0
\(238\) −0.0431181 −0.00279493
\(239\) −1.00000 −0.0646846
\(240\) 0 0
\(241\) 14.9582 0.963542 0.481771 0.876297i \(-0.339994\pi\)
0.481771 + 0.876297i \(0.339994\pi\)
\(242\) −0.556744 −0.0357889
\(243\) 0 0
\(244\) 26.0614 1.66841
\(245\) 29.5485 1.88778
\(246\) 0 0
\(247\) 6.79535 0.432378
\(248\) −7.60690 −0.483039
\(249\) 0 0
\(250\) 8.09959 0.512263
\(251\) −15.6861 −0.990098 −0.495049 0.868865i \(-0.664850\pi\)
−0.495049 + 0.868865i \(0.664850\pi\)
\(252\) 0 0
\(253\) −4.85075 −0.304964
\(254\) 2.54673 0.159796
\(255\) 0 0
\(256\) 11.4354 0.714715
\(257\) −25.8292 −1.61118 −0.805590 0.592474i \(-0.798151\pi\)
−0.805590 + 0.592474i \(0.798151\pi\)
\(258\) 0 0
\(259\) −0.215513 −0.0133913
\(260\) 52.1678 3.23531
\(261\) 0 0
\(262\) −0.432607 −0.0267266
\(263\) 19.4124 1.19702 0.598511 0.801115i \(-0.295759\pi\)
0.598511 + 0.801115i \(0.295759\pi\)
\(264\) 0 0
\(265\) 41.1819 2.52979
\(266\) −0.0195384 −0.00119798
\(267\) 0 0
\(268\) 8.27853 0.505692
\(269\) −0.0307895 −0.00187727 −0.000938635 1.00000i \(-0.500299\pi\)
−0.000938635 1.00000i \(0.500299\pi\)
\(270\) 0 0
\(271\) −7.27426 −0.441880 −0.220940 0.975287i \(-0.570912\pi\)
−0.220940 + 0.975287i \(0.570912\pi\)
\(272\) 8.58978 0.520832
\(273\) 0 0
\(274\) 1.67644 0.101277
\(275\) −46.8154 −2.82307
\(276\) 0 0
\(277\) 2.12921 0.127932 0.0639659 0.997952i \(-0.479625\pi\)
0.0639659 + 0.997952i \(0.479625\pi\)
\(278\) −2.16415 −0.129797
\(279\) 0 0
\(280\) −0.304607 −0.0182038
\(281\) 12.3210 0.735007 0.367504 0.930022i \(-0.380213\pi\)
0.367504 + 0.930022i \(0.380213\pi\)
\(282\) 0 0
\(283\) −20.6645 −1.22838 −0.614189 0.789159i \(-0.710517\pi\)
−0.614189 + 0.789159i \(0.710517\pi\)
\(284\) −27.2250 −1.61551
\(285\) 0 0
\(286\) 5.66610 0.335043
\(287\) −0.533684 −0.0315024
\(288\) 0 0
\(289\) −11.4477 −0.673395
\(290\) 8.52269 0.500469
\(291\) 0 0
\(292\) −12.9267 −0.756476
\(293\) −25.1094 −1.46691 −0.733454 0.679740i \(-0.762093\pi\)
−0.733454 + 0.679740i \(0.762093\pi\)
\(294\) 0 0
\(295\) −18.3394 −1.06776
\(296\) −2.77011 −0.161010
\(297\) 0 0
\(298\) 2.03478 0.117872
\(299\) 8.47181 0.489938
\(300\) 0 0
\(301\) 0.471150 0.0271566
\(302\) 5.85757 0.337065
\(303\) 0 0
\(304\) 3.89235 0.223242
\(305\) 56.7431 3.24910
\(306\) 0 0
\(307\) 14.4578 0.825151 0.412575 0.910924i \(-0.364629\pi\)
0.412575 + 0.910924i \(0.364629\pi\)
\(308\) 0.529558 0.0301744
\(309\) 0 0
\(310\) −8.15574 −0.463215
\(311\) 26.3144 1.49215 0.746075 0.665861i \(-0.231936\pi\)
0.746075 + 0.665861i \(0.231936\pi\)
\(312\) 0 0
\(313\) −11.3681 −0.642563 −0.321281 0.946984i \(-0.604114\pi\)
−0.321281 + 0.946984i \(0.604114\pi\)
\(314\) 1.72976 0.0976161
\(315\) 0 0
\(316\) −33.9875 −1.91195
\(317\) 21.3000 1.19633 0.598163 0.801374i \(-0.295897\pi\)
0.598163 + 0.801374i \(0.295897\pi\)
\(318\) 0 0
\(319\) −30.0891 −1.68467
\(320\) 27.8942 1.55933
\(321\) 0 0
\(322\) −0.0243587 −0.00135746
\(323\) 2.51595 0.139991
\(324\) 0 0
\(325\) 81.7629 4.53539
\(326\) 1.39971 0.0775230
\(327\) 0 0
\(328\) −6.85977 −0.378767
\(329\) −0.505143 −0.0278494
\(330\) 0 0
\(331\) 6.61971 0.363852 0.181926 0.983312i \(-0.441767\pi\)
0.181926 + 0.983312i \(0.441767\pi\)
\(332\) 21.4641 1.17799
\(333\) 0 0
\(334\) −2.85218 −0.156064
\(335\) 18.0247 0.984795
\(336\) 0 0
\(337\) −10.0337 −0.546572 −0.273286 0.961933i \(-0.588110\pi\)
−0.273286 + 0.961933i \(0.588110\pi\)
\(338\) −6.71965 −0.365501
\(339\) 0 0
\(340\) 19.3149 1.04750
\(341\) 28.7937 1.55926
\(342\) 0 0
\(343\) −1.04814 −0.0565942
\(344\) 6.05598 0.326516
\(345\) 0 0
\(346\) −0.977474 −0.0525493
\(347\) −3.07598 −0.165127 −0.0825636 0.996586i \(-0.526311\pi\)
−0.0825636 + 0.996586i \(0.526311\pi\)
\(348\) 0 0
\(349\) −6.65984 −0.356493 −0.178247 0.983986i \(-0.557042\pi\)
−0.178247 + 0.983986i \(0.557042\pi\)
\(350\) −0.235090 −0.0125661
\(351\) 0 0
\(352\) 10.2617 0.546948
\(353\) 0.692908 0.0368798 0.0184399 0.999830i \(-0.494130\pi\)
0.0184399 + 0.999830i \(0.494130\pi\)
\(354\) 0 0
\(355\) −59.2766 −3.14608
\(356\) 20.5152 1.08730
\(357\) 0 0
\(358\) 4.41930 0.233567
\(359\) 2.40470 0.126915 0.0634576 0.997985i \(-0.479787\pi\)
0.0634576 + 0.997985i \(0.479787\pi\)
\(360\) 0 0
\(361\) −17.8599 −0.939996
\(362\) −2.95292 −0.155202
\(363\) 0 0
\(364\) −0.924871 −0.0484764
\(365\) −28.1450 −1.47318
\(366\) 0 0
\(367\) −30.6458 −1.59970 −0.799848 0.600202i \(-0.795087\pi\)
−0.799848 + 0.600202i \(0.795087\pi\)
\(368\) 4.85262 0.252960
\(369\) 0 0
\(370\) −2.96998 −0.154402
\(371\) −0.730106 −0.0379052
\(372\) 0 0
\(373\) −25.7277 −1.33213 −0.666065 0.745894i \(-0.732023\pi\)
−0.666065 + 0.745894i \(0.732023\pi\)
\(374\) 2.09785 0.108477
\(375\) 0 0
\(376\) −6.49291 −0.334846
\(377\) 52.5505 2.70649
\(378\) 0 0
\(379\) −28.5899 −1.46856 −0.734281 0.678845i \(-0.762481\pi\)
−0.734281 + 0.678845i \(0.762481\pi\)
\(380\) 8.75229 0.448983
\(381\) 0 0
\(382\) −2.07587 −0.106211
\(383\) −5.30492 −0.271069 −0.135534 0.990773i \(-0.543275\pi\)
−0.135534 + 0.990773i \(0.543275\pi\)
\(384\) 0 0
\(385\) 1.15300 0.0587623
\(386\) 3.61933 0.184219
\(387\) 0 0
\(388\) −18.6667 −0.947656
\(389\) −7.61972 −0.386335 −0.193168 0.981166i \(-0.561876\pi\)
−0.193168 + 0.981166i \(0.561876\pi\)
\(390\) 0 0
\(391\) 3.13665 0.158627
\(392\) −6.73348 −0.340092
\(393\) 0 0
\(394\) 0.770297 0.0388070
\(395\) −74.0004 −3.72336
\(396\) 0 0
\(397\) 6.58074 0.330278 0.165139 0.986270i \(-0.447193\pi\)
0.165139 + 0.986270i \(0.447193\pi\)
\(398\) 3.92607 0.196796
\(399\) 0 0
\(400\) 46.8335 2.34167
\(401\) −10.2473 −0.511727 −0.255864 0.966713i \(-0.582360\pi\)
−0.255864 + 0.966713i \(0.582360\pi\)
\(402\) 0 0
\(403\) −50.2880 −2.50502
\(404\) −15.5791 −0.775091
\(405\) 0 0
\(406\) −0.151097 −0.00749881
\(407\) 10.4854 0.519744
\(408\) 0 0
\(409\) 4.75187 0.234965 0.117483 0.993075i \(-0.462518\pi\)
0.117483 + 0.993075i \(0.462518\pi\)
\(410\) −7.35470 −0.363223
\(411\) 0 0
\(412\) −3.06196 −0.150852
\(413\) 0.325135 0.0159988
\(414\) 0 0
\(415\) 46.7334 2.29405
\(416\) −17.9219 −0.878695
\(417\) 0 0
\(418\) 0.950613 0.0464960
\(419\) 15.8353 0.773605 0.386803 0.922163i \(-0.373579\pi\)
0.386803 + 0.922163i \(0.373579\pi\)
\(420\) 0 0
\(421\) −20.4926 −0.998749 −0.499375 0.866386i \(-0.666437\pi\)
−0.499375 + 0.866386i \(0.666437\pi\)
\(422\) −4.11365 −0.200250
\(423\) 0 0
\(424\) −9.38449 −0.455751
\(425\) 30.2723 1.46842
\(426\) 0 0
\(427\) −1.00599 −0.0486831
\(428\) −34.6439 −1.67458
\(429\) 0 0
\(430\) 6.49291 0.313116
\(431\) −4.25692 −0.205049 −0.102524 0.994731i \(-0.532692\pi\)
−0.102524 + 0.994731i \(0.532692\pi\)
\(432\) 0 0
\(433\) −24.2092 −1.16342 −0.581710 0.813396i \(-0.697616\pi\)
−0.581710 + 0.813396i \(0.697616\pi\)
\(434\) 0.144591 0.00694061
\(435\) 0 0
\(436\) −7.14736 −0.342297
\(437\) 1.42133 0.0679916
\(438\) 0 0
\(439\) −26.7197 −1.27526 −0.637630 0.770343i \(-0.720085\pi\)
−0.637630 + 0.770343i \(0.720085\pi\)
\(440\) 14.8202 0.706525
\(441\) 0 0
\(442\) −3.66388 −0.174273
\(443\) 17.3125 0.822542 0.411271 0.911513i \(-0.365085\pi\)
0.411271 + 0.911513i \(0.365085\pi\)
\(444\) 0 0
\(445\) 44.6674 2.11744
\(446\) 5.26477 0.249294
\(447\) 0 0
\(448\) −0.494530 −0.0233644
\(449\) −28.6843 −1.35369 −0.676847 0.736124i \(-0.736654\pi\)
−0.676847 + 0.736124i \(0.736654\pi\)
\(450\) 0 0
\(451\) 25.9656 1.22267
\(452\) 34.5595 1.62554
\(453\) 0 0
\(454\) 4.18833 0.196568
\(455\) −2.01371 −0.0944040
\(456\) 0 0
\(457\) −20.0915 −0.939841 −0.469920 0.882709i \(-0.655717\pi\)
−0.469920 + 0.882709i \(0.655717\pi\)
\(458\) 3.18316 0.148739
\(459\) 0 0
\(460\) 10.9116 0.508754
\(461\) −1.64319 −0.0765311 −0.0382655 0.999268i \(-0.512183\pi\)
−0.0382655 + 0.999268i \(0.512183\pi\)
\(462\) 0 0
\(463\) 13.3260 0.619310 0.309655 0.950849i \(-0.399786\pi\)
0.309655 + 0.950849i \(0.399786\pi\)
\(464\) 30.1008 1.39739
\(465\) 0 0
\(466\) 4.41066 0.204320
\(467\) 3.59055 0.166151 0.0830755 0.996543i \(-0.473526\pi\)
0.0830755 + 0.996543i \(0.473526\pi\)
\(468\) 0 0
\(469\) −0.319556 −0.0147557
\(470\) −6.96137 −0.321104
\(471\) 0 0
\(472\) 4.17916 0.192361
\(473\) −22.9231 −1.05400
\(474\) 0 0
\(475\) 13.7175 0.629403
\(476\) −0.342429 −0.0156952
\(477\) 0 0
\(478\) 0.244320 0.0111750
\(479\) 28.7910 1.31549 0.657747 0.753239i \(-0.271510\pi\)
0.657747 + 0.753239i \(0.271510\pi\)
\(480\) 0 0
\(481\) −18.3128 −0.834990
\(482\) −3.65459 −0.166462
\(483\) 0 0
\(484\) −4.42147 −0.200976
\(485\) −40.6426 −1.84549
\(486\) 0 0
\(487\) 8.43245 0.382111 0.191055 0.981579i \(-0.438809\pi\)
0.191055 + 0.981579i \(0.438809\pi\)
\(488\) −12.9306 −0.585339
\(489\) 0 0
\(490\) −7.21930 −0.326135
\(491\) 4.08991 0.184575 0.0922875 0.995732i \(-0.470582\pi\)
0.0922875 + 0.995732i \(0.470582\pi\)
\(492\) 0 0
\(493\) 19.4566 0.876281
\(494\) −1.66024 −0.0746978
\(495\) 0 0
\(496\) −28.8048 −1.29337
\(497\) 1.05090 0.0471394
\(498\) 0 0
\(499\) −6.56133 −0.293725 −0.146863 0.989157i \(-0.546918\pi\)
−0.146863 + 0.989157i \(0.546918\pi\)
\(500\) 64.3241 2.87666
\(501\) 0 0
\(502\) 3.83243 0.171050
\(503\) 9.32533 0.415796 0.207898 0.978151i \(-0.433338\pi\)
0.207898 + 0.978151i \(0.433338\pi\)
\(504\) 0 0
\(505\) −33.9202 −1.50943
\(506\) 1.18514 0.0526858
\(507\) 0 0
\(508\) 20.2253 0.897352
\(509\) 8.28590 0.367266 0.183633 0.982995i \(-0.441214\pi\)
0.183633 + 0.982995i \(0.441214\pi\)
\(510\) 0 0
\(511\) 0.498977 0.0220735
\(512\) −17.2845 −0.763873
\(513\) 0 0
\(514\) 6.31059 0.278348
\(515\) −6.66676 −0.293773
\(516\) 0 0
\(517\) 24.5770 1.08089
\(518\) 0.0526541 0.00231349
\(519\) 0 0
\(520\) −25.8834 −1.13506
\(521\) 0.470820 0.0206270 0.0103135 0.999947i \(-0.496717\pi\)
0.0103135 + 0.999947i \(0.496717\pi\)
\(522\) 0 0
\(523\) 3.00357 0.131337 0.0656685 0.997841i \(-0.479082\pi\)
0.0656685 + 0.997841i \(0.479082\pi\)
\(524\) −3.43562 −0.150086
\(525\) 0 0
\(526\) −4.74285 −0.206798
\(527\) −18.6189 −0.811052
\(528\) 0 0
\(529\) −21.2280 −0.922957
\(530\) −10.0616 −0.437047
\(531\) 0 0
\(532\) −0.155168 −0.00672737
\(533\) −45.3488 −1.96427
\(534\) 0 0
\(535\) −75.4297 −3.26111
\(536\) −4.10745 −0.177415
\(537\) 0 0
\(538\) 0.00752251 0.000324318 0
\(539\) 25.4876 1.09783
\(540\) 0 0
\(541\) −1.18196 −0.0508164 −0.0254082 0.999677i \(-0.508089\pi\)
−0.0254082 + 0.999677i \(0.508089\pi\)
\(542\) 1.77725 0.0763394
\(543\) 0 0
\(544\) −6.63552 −0.284495
\(545\) −15.5618 −0.666596
\(546\) 0 0
\(547\) 32.8349 1.40392 0.701960 0.712217i \(-0.252309\pi\)
0.701960 + 0.712217i \(0.252309\pi\)
\(548\) 13.3137 0.568733
\(549\) 0 0
\(550\) 11.4380 0.487716
\(551\) 8.81651 0.375596
\(552\) 0 0
\(553\) 1.31194 0.0557893
\(554\) −0.520209 −0.0221016
\(555\) 0 0
\(556\) −17.1869 −0.728888
\(557\) −15.9291 −0.674938 −0.337469 0.941337i \(-0.609571\pi\)
−0.337469 + 0.941337i \(0.609571\pi\)
\(558\) 0 0
\(559\) 40.0350 1.69330
\(560\) −1.15344 −0.0487419
\(561\) 0 0
\(562\) −3.01026 −0.126980
\(563\) 37.8619 1.59569 0.797844 0.602864i \(-0.205974\pi\)
0.797844 + 0.602864i \(0.205974\pi\)
\(564\) 0 0
\(565\) 75.2458 3.16561
\(566\) 5.04876 0.212215
\(567\) 0 0
\(568\) 13.5079 0.566778
\(569\) 25.3083 1.06098 0.530489 0.847692i \(-0.322008\pi\)
0.530489 + 0.847692i \(0.322008\pi\)
\(570\) 0 0
\(571\) 10.3713 0.434027 0.217013 0.976169i \(-0.430369\pi\)
0.217013 + 0.976169i \(0.430369\pi\)
\(572\) 44.9982 1.88147
\(573\) 0 0
\(574\) 0.130390 0.00544237
\(575\) 17.1018 0.713192
\(576\) 0 0
\(577\) −31.2506 −1.30098 −0.650490 0.759515i \(-0.725436\pi\)
−0.650490 + 0.759515i \(0.725436\pi\)
\(578\) 2.79691 0.116336
\(579\) 0 0
\(580\) 67.6842 2.81043
\(581\) −0.828526 −0.0343731
\(582\) 0 0
\(583\) 35.5222 1.47118
\(584\) 6.41366 0.265399
\(585\) 0 0
\(586\) 6.13474 0.253424
\(587\) 34.9885 1.44413 0.722066 0.691825i \(-0.243193\pi\)
0.722066 + 0.691825i \(0.243193\pi\)
\(588\) 0 0
\(589\) −8.43692 −0.347637
\(590\) 4.48068 0.184467
\(591\) 0 0
\(592\) −10.4895 −0.431115
\(593\) −1.74281 −0.0715688 −0.0357844 0.999360i \(-0.511393\pi\)
−0.0357844 + 0.999360i \(0.511393\pi\)
\(594\) 0 0
\(595\) −0.745566 −0.0305652
\(596\) 16.1595 0.661919
\(597\) 0 0
\(598\) −2.06984 −0.0846419
\(599\) −26.0525 −1.06448 −0.532238 0.846595i \(-0.678649\pi\)
−0.532238 + 0.846595i \(0.678649\pi\)
\(600\) 0 0
\(601\) −22.3167 −0.910318 −0.455159 0.890410i \(-0.650418\pi\)
−0.455159 + 0.890410i \(0.650418\pi\)
\(602\) −0.115112 −0.00469160
\(603\) 0 0
\(604\) 46.5188 1.89282
\(605\) −9.62679 −0.391385
\(606\) 0 0
\(607\) 40.8318 1.65731 0.828655 0.559760i \(-0.189107\pi\)
0.828655 + 0.559760i \(0.189107\pi\)
\(608\) −3.00680 −0.121942
\(609\) 0 0
\(610\) −13.8635 −0.561317
\(611\) −42.9235 −1.73650
\(612\) 0 0
\(613\) 19.6260 0.792687 0.396344 0.918102i \(-0.370279\pi\)
0.396344 + 0.918102i \(0.370279\pi\)
\(614\) −3.53234 −0.142553
\(615\) 0 0
\(616\) −0.262744 −0.0105863
\(617\) −16.8830 −0.679683 −0.339841 0.940483i \(-0.610373\pi\)
−0.339841 + 0.940483i \(0.610373\pi\)
\(618\) 0 0
\(619\) 21.3216 0.856986 0.428493 0.903545i \(-0.359045\pi\)
0.428493 + 0.903545i \(0.359045\pi\)
\(620\) −64.7701 −2.60123
\(621\) 0 0
\(622\) −6.42913 −0.257785
\(623\) −0.791898 −0.0317267
\(624\) 0 0
\(625\) 75.8156 3.03262
\(626\) 2.77746 0.111010
\(627\) 0 0
\(628\) 13.7372 0.548173
\(629\) −6.78021 −0.270345
\(630\) 0 0
\(631\) 28.2400 1.12422 0.562109 0.827063i \(-0.309990\pi\)
0.562109 + 0.827063i \(0.309990\pi\)
\(632\) 16.8631 0.670779
\(633\) 0 0
\(634\) −5.20402 −0.206678
\(635\) 44.0362 1.74752
\(636\) 0 0
\(637\) −44.5139 −1.76371
\(638\) 7.35139 0.291044
\(639\) 0 0
\(640\) −30.6084 −1.20990
\(641\) 33.9514 1.34100 0.670500 0.741910i \(-0.266080\pi\)
0.670500 + 0.741910i \(0.266080\pi\)
\(642\) 0 0
\(643\) −27.1751 −1.07168 −0.535841 0.844319i \(-0.680005\pi\)
−0.535841 + 0.844319i \(0.680005\pi\)
\(644\) −0.193449 −0.00762294
\(645\) 0 0
\(646\) −0.614697 −0.0241849
\(647\) 42.6110 1.67521 0.837606 0.546275i \(-0.183955\pi\)
0.837606 + 0.546275i \(0.183955\pi\)
\(648\) 0 0
\(649\) −15.8189 −0.620948
\(650\) −19.9763 −0.783536
\(651\) 0 0
\(652\) 11.1160 0.435338
\(653\) −19.5670 −0.765715 −0.382857 0.923807i \(-0.625060\pi\)
−0.382857 + 0.923807i \(0.625060\pi\)
\(654\) 0 0
\(655\) −7.48031 −0.292280
\(656\) −25.9756 −1.01418
\(657\) 0 0
\(658\) 0.123417 0.00481128
\(659\) −24.6924 −0.961878 −0.480939 0.876754i \(-0.659704\pi\)
−0.480939 + 0.876754i \(0.659704\pi\)
\(660\) 0 0
\(661\) 31.9176 1.24145 0.620725 0.784028i \(-0.286838\pi\)
0.620725 + 0.784028i \(0.286838\pi\)
\(662\) −1.61733 −0.0628593
\(663\) 0 0
\(664\) −10.6495 −0.413283
\(665\) −0.337844 −0.0131010
\(666\) 0 0
\(667\) 10.9916 0.425597
\(668\) −22.6510 −0.876394
\(669\) 0 0
\(670\) −4.40380 −0.170134
\(671\) 48.9448 1.88949
\(672\) 0 0
\(673\) 13.7390 0.529599 0.264799 0.964303i \(-0.414694\pi\)
0.264799 + 0.964303i \(0.414694\pi\)
\(674\) 2.45144 0.0944260
\(675\) 0 0
\(676\) −53.3651 −2.05250
\(677\) −0.981890 −0.0377371 −0.0188685 0.999822i \(-0.506006\pi\)
−0.0188685 + 0.999822i \(0.506006\pi\)
\(678\) 0 0
\(679\) 0.720545 0.0276520
\(680\) −9.58321 −0.367499
\(681\) 0 0
\(682\) −7.03487 −0.269379
\(683\) 19.2274 0.735716 0.367858 0.929882i \(-0.380091\pi\)
0.367858 + 0.929882i \(0.380091\pi\)
\(684\) 0 0
\(685\) 28.9877 1.10756
\(686\) 0.256082 0.00977724
\(687\) 0 0
\(688\) 22.9319 0.874272
\(689\) −62.0393 −2.36351
\(690\) 0 0
\(691\) −27.3404 −1.04008 −0.520038 0.854143i \(-0.674082\pi\)
−0.520038 + 0.854143i \(0.674082\pi\)
\(692\) −7.76276 −0.295096
\(693\) 0 0
\(694\) 0.751524 0.0285275
\(695\) −37.4208 −1.41945
\(696\) 0 0
\(697\) −16.7902 −0.635973
\(698\) 1.62714 0.0615880
\(699\) 0 0
\(700\) −1.86701 −0.0705662
\(701\) −4.84865 −0.183131 −0.0915654 0.995799i \(-0.529187\pi\)
−0.0915654 + 0.995799i \(0.529187\pi\)
\(702\) 0 0
\(703\) −3.07237 −0.115877
\(704\) 24.0606 0.906819
\(705\) 0 0
\(706\) −0.169291 −0.00637137
\(707\) 0.601364 0.0226166
\(708\) 0 0
\(709\) 42.1926 1.58457 0.792287 0.610148i \(-0.208890\pi\)
0.792287 + 0.610148i \(0.208890\pi\)
\(710\) 14.4825 0.543518
\(711\) 0 0
\(712\) −10.1787 −0.381465
\(713\) −10.5184 −0.393916
\(714\) 0 0
\(715\) 97.9738 3.66401
\(716\) 35.0965 1.31162
\(717\) 0 0
\(718\) −0.587517 −0.0219259
\(719\) −33.7164 −1.25741 −0.628704 0.777645i \(-0.716414\pi\)
−0.628704 + 0.777645i \(0.716414\pi\)
\(720\) 0 0
\(721\) 0.118194 0.00440176
\(722\) 4.36354 0.162394
\(723\) 0 0
\(724\) −23.4511 −0.871553
\(725\) 106.082 3.93978
\(726\) 0 0
\(727\) −15.2627 −0.566061 −0.283030 0.959111i \(-0.591340\pi\)
−0.283030 + 0.959111i \(0.591340\pi\)
\(728\) 0.458881 0.0170073
\(729\) 0 0
\(730\) 6.87640 0.254507
\(731\) 14.8228 0.548241
\(732\) 0 0
\(733\) 1.87172 0.0691336 0.0345668 0.999402i \(-0.488995\pi\)
0.0345668 + 0.999402i \(0.488995\pi\)
\(734\) 7.48739 0.276364
\(735\) 0 0
\(736\) −3.74860 −0.138175
\(737\) 15.5475 0.572700
\(738\) 0 0
\(739\) 11.6680 0.429214 0.214607 0.976701i \(-0.431153\pi\)
0.214607 + 0.976701i \(0.431153\pi\)
\(740\) −23.5865 −0.867058
\(741\) 0 0
\(742\) 0.178380 0.00654852
\(743\) −14.6237 −0.536491 −0.268245 0.963351i \(-0.586444\pi\)
−0.268245 + 0.963351i \(0.586444\pi\)
\(744\) 0 0
\(745\) 35.1838 1.28904
\(746\) 6.28580 0.230140
\(747\) 0 0
\(748\) 16.6604 0.609164
\(749\) 1.33728 0.0488630
\(750\) 0 0
\(751\) −47.4417 −1.73117 −0.865586 0.500761i \(-0.833054\pi\)
−0.865586 + 0.500761i \(0.833054\pi\)
\(752\) −24.5864 −0.896575
\(753\) 0 0
\(754\) −12.8392 −0.467575
\(755\) 101.285 3.68612
\(756\) 0 0
\(757\) 32.1954 1.17016 0.585082 0.810974i \(-0.301062\pi\)
0.585082 + 0.810974i \(0.301062\pi\)
\(758\) 6.98508 0.253710
\(759\) 0 0
\(760\) −4.34251 −0.157519
\(761\) −25.0702 −0.908794 −0.454397 0.890799i \(-0.650145\pi\)
−0.454397 + 0.890799i \(0.650145\pi\)
\(762\) 0 0
\(763\) 0.275893 0.00998798
\(764\) −16.4858 −0.596436
\(765\) 0 0
\(766\) 1.29610 0.0468300
\(767\) 27.6277 0.997579
\(768\) 0 0
\(769\) 7.38439 0.266288 0.133144 0.991097i \(-0.457493\pi\)
0.133144 + 0.991097i \(0.457493\pi\)
\(770\) −0.281701 −0.0101518
\(771\) 0 0
\(772\) 28.7435 1.03450
\(773\) −6.55568 −0.235792 −0.117896 0.993026i \(-0.537615\pi\)
−0.117896 + 0.993026i \(0.537615\pi\)
\(774\) 0 0
\(775\) −101.515 −3.64651
\(776\) 9.26160 0.332472
\(777\) 0 0
\(778\) 1.86165 0.0667435
\(779\) −7.60826 −0.272594
\(780\) 0 0
\(781\) −51.1300 −1.82958
\(782\) −0.766347 −0.0274045
\(783\) 0 0
\(784\) −25.4974 −0.910622
\(785\) 29.9097 1.06752
\(786\) 0 0
\(787\) −37.9638 −1.35326 −0.676632 0.736321i \(-0.736561\pi\)
−0.676632 + 0.736321i \(0.736561\pi\)
\(788\) 6.11743 0.217924
\(789\) 0 0
\(790\) 18.0798 0.643250
\(791\) −1.33402 −0.0474322
\(792\) 0 0
\(793\) −85.4818 −3.03555
\(794\) −1.60781 −0.0570590
\(795\) 0 0
\(796\) 31.1795 1.10513
\(797\) 7.47891 0.264917 0.132458 0.991189i \(-0.457713\pi\)
0.132458 + 0.991189i \(0.457713\pi\)
\(798\) 0 0
\(799\) −15.8922 −0.562227
\(800\) −36.1784 −1.27910
\(801\) 0 0
\(802\) 2.50363 0.0884062
\(803\) −24.2770 −0.856716
\(804\) 0 0
\(805\) −0.421193 −0.0148451
\(806\) 12.2864 0.432769
\(807\) 0 0
\(808\) 7.72970 0.271930
\(809\) 19.1096 0.671858 0.335929 0.941887i \(-0.390950\pi\)
0.335929 + 0.941887i \(0.390950\pi\)
\(810\) 0 0
\(811\) 1.33294 0.0468057 0.0234028 0.999726i \(-0.492550\pi\)
0.0234028 + 0.999726i \(0.492550\pi\)
\(812\) −1.19996 −0.0421103
\(813\) 0 0
\(814\) −2.56180 −0.0897912
\(815\) 24.2028 0.847787
\(816\) 0 0
\(817\) 6.71676 0.234990
\(818\) −1.16098 −0.0405927
\(819\) 0 0
\(820\) −58.4085 −2.03971
\(821\) 27.6681 0.965623 0.482812 0.875724i \(-0.339616\pi\)
0.482812 + 0.875724i \(0.339616\pi\)
\(822\) 0 0
\(823\) −14.1900 −0.494634 −0.247317 0.968935i \(-0.579549\pi\)
−0.247317 + 0.968935i \(0.579549\pi\)
\(824\) 1.51921 0.0529243
\(825\) 0 0
\(826\) −0.0794371 −0.00276397
\(827\) −20.9131 −0.727220 −0.363610 0.931551i \(-0.618456\pi\)
−0.363610 + 0.931551i \(0.618456\pi\)
\(828\) 0 0
\(829\) −11.6318 −0.403989 −0.201995 0.979387i \(-0.564742\pi\)
−0.201995 + 0.979387i \(0.564742\pi\)
\(830\) −11.4179 −0.396321
\(831\) 0 0
\(832\) −42.0217 −1.45684
\(833\) −16.4811 −0.571035
\(834\) 0 0
\(835\) −49.3177 −1.70671
\(836\) 7.54944 0.261103
\(837\) 0 0
\(838\) −3.86889 −0.133648
\(839\) 11.9369 0.412107 0.206053 0.978541i \(-0.433938\pi\)
0.206053 + 0.978541i \(0.433938\pi\)
\(840\) 0 0
\(841\) 39.1809 1.35106
\(842\) 5.00676 0.172544
\(843\) 0 0
\(844\) −32.6692 −1.12452
\(845\) −116.191 −3.99709
\(846\) 0 0
\(847\) 0.170671 0.00586434
\(848\) −35.5359 −1.22031
\(849\) 0 0
\(850\) −7.39614 −0.253686
\(851\) −3.83035 −0.131303
\(852\) 0 0
\(853\) −19.2324 −0.658506 −0.329253 0.944242i \(-0.606797\pi\)
−0.329253 + 0.944242i \(0.606797\pi\)
\(854\) 0.245783 0.00841053
\(855\) 0 0
\(856\) 17.1888 0.587502
\(857\) 9.11993 0.311531 0.155765 0.987794i \(-0.450216\pi\)
0.155765 + 0.987794i \(0.450216\pi\)
\(858\) 0 0
\(859\) 41.6196 1.42004 0.710022 0.704180i \(-0.248685\pi\)
0.710022 + 0.704180i \(0.248685\pi\)
\(860\) 51.5645 1.75833
\(861\) 0 0
\(862\) 1.04005 0.0354243
\(863\) −30.3148 −1.03193 −0.515964 0.856610i \(-0.672566\pi\)
−0.515964 + 0.856610i \(0.672566\pi\)
\(864\) 0 0
\(865\) −16.9017 −0.574676
\(866\) 5.91480 0.200993
\(867\) 0 0
\(868\) 1.14830 0.0389757
\(869\) −63.8303 −2.16529
\(870\) 0 0
\(871\) −27.1537 −0.920067
\(872\) 3.54621 0.120090
\(873\) 0 0
\(874\) −0.347261 −0.0117463
\(875\) −2.48295 −0.0839390
\(876\) 0 0
\(877\) 34.0049 1.14826 0.574131 0.818763i \(-0.305340\pi\)
0.574131 + 0.818763i \(0.305340\pi\)
\(878\) 6.52815 0.220315
\(879\) 0 0
\(880\) 56.1191 1.89177
\(881\) −33.5155 −1.12917 −0.564584 0.825376i \(-0.690963\pi\)
−0.564584 + 0.825376i \(0.690963\pi\)
\(882\) 0 0
\(883\) 3.58518 0.120651 0.0603255 0.998179i \(-0.480786\pi\)
0.0603255 + 0.998179i \(0.480786\pi\)
\(884\) −29.0973 −0.978647
\(885\) 0 0
\(886\) −4.22980 −0.142103
\(887\) −43.2091 −1.45082 −0.725410 0.688317i \(-0.758350\pi\)
−0.725410 + 0.688317i \(0.758350\pi\)
\(888\) 0 0
\(889\) −0.780708 −0.0261841
\(890\) −10.9131 −0.365809
\(891\) 0 0
\(892\) 41.8110 1.39994
\(893\) −7.20137 −0.240985
\(894\) 0 0
\(895\) 76.4151 2.55428
\(896\) 0.542650 0.0181287
\(897\) 0 0
\(898\) 7.00815 0.233865
\(899\) −65.2453 −2.17605
\(900\) 0 0
\(901\) −22.9698 −0.765234
\(902\) −6.34392 −0.211230
\(903\) 0 0
\(904\) −17.1469 −0.570298
\(905\) −51.0597 −1.69728
\(906\) 0 0
\(907\) 0.941818 0.0312726 0.0156363 0.999878i \(-0.495023\pi\)
0.0156363 + 0.999878i \(0.495023\pi\)
\(908\) 33.2622 1.10385
\(909\) 0 0
\(910\) 0.491990 0.0163093
\(911\) −21.2909 −0.705398 −0.352699 0.935737i \(-0.614736\pi\)
−0.352699 + 0.935737i \(0.614736\pi\)
\(912\) 0 0
\(913\) 40.3107 1.33409
\(914\) 4.90876 0.162367
\(915\) 0 0
\(916\) 25.2795 0.835260
\(917\) 0.132617 0.00437940
\(918\) 0 0
\(919\) 3.97921 0.131262 0.0656310 0.997844i \(-0.479094\pi\)
0.0656310 + 0.997844i \(0.479094\pi\)
\(920\) −5.41384 −0.178489
\(921\) 0 0
\(922\) 0.401465 0.0132215
\(923\) 89.2984 2.93929
\(924\) 0 0
\(925\) −36.9673 −1.21548
\(926\) −3.25580 −0.106992
\(927\) 0 0
\(928\) −23.2525 −0.763302
\(929\) −24.9145 −0.817417 −0.408709 0.912665i \(-0.634021\pi\)
−0.408709 + 0.912665i \(0.634021\pi\)
\(930\) 0 0
\(931\) −7.46819 −0.244760
\(932\) 35.0279 1.14738
\(933\) 0 0
\(934\) −0.877245 −0.0287043
\(935\) 36.2744 1.18630
\(936\) 0 0
\(937\) −7.37922 −0.241069 −0.120534 0.992709i \(-0.538461\pi\)
−0.120534 + 0.992709i \(0.538461\pi\)
\(938\) 0.0780741 0.00254921
\(939\) 0 0
\(940\) −55.2848 −1.80319
\(941\) 19.7016 0.642255 0.321127 0.947036i \(-0.395938\pi\)
0.321127 + 0.947036i \(0.395938\pi\)
\(942\) 0 0
\(943\) −9.48528 −0.308883
\(944\) 15.8251 0.515062
\(945\) 0 0
\(946\) 5.60057 0.182090
\(947\) −3.62801 −0.117895 −0.0589473 0.998261i \(-0.518774\pi\)
−0.0589473 + 0.998261i \(0.518774\pi\)
\(948\) 0 0
\(949\) 42.3996 1.37635
\(950\) −3.35147 −0.108736
\(951\) 0 0
\(952\) 0.169899 0.00550645
\(953\) 7.87411 0.255067 0.127534 0.991834i \(-0.459294\pi\)
0.127534 + 0.991834i \(0.459294\pi\)
\(954\) 0 0
\(955\) −35.8943 −1.16151
\(956\) 1.94031 0.0627540
\(957\) 0 0
\(958\) −7.03422 −0.227265
\(959\) −0.513917 −0.0165952
\(960\) 0 0
\(961\) 31.4362 1.01407
\(962\) 4.47418 0.144253
\(963\) 0 0
\(964\) −29.0235 −0.934784
\(965\) 62.5828 2.01461
\(966\) 0 0
\(967\) −21.9578 −0.706116 −0.353058 0.935601i \(-0.614858\pi\)
−0.353058 + 0.935601i \(0.614858\pi\)
\(968\) 2.19374 0.0705096
\(969\) 0 0
\(970\) 9.92982 0.318827
\(971\) 26.9544 0.865007 0.432504 0.901632i \(-0.357630\pi\)
0.432504 + 0.901632i \(0.357630\pi\)
\(972\) 0 0
\(973\) 0.663426 0.0212685
\(974\) −2.06022 −0.0660137
\(975\) 0 0
\(976\) −48.9637 −1.56729
\(977\) −25.4387 −0.813855 −0.406927 0.913460i \(-0.633400\pi\)
−0.406927 + 0.913460i \(0.633400\pi\)
\(978\) 0 0
\(979\) 38.5286 1.23138
\(980\) −57.3332 −1.83144
\(981\) 0 0
\(982\) −0.999247 −0.0318873
\(983\) 3.71985 0.118645 0.0593224 0.998239i \(-0.481106\pi\)
0.0593224 + 0.998239i \(0.481106\pi\)
\(984\) 0 0
\(985\) 13.3194 0.424391
\(986\) −4.75364 −0.151387
\(987\) 0 0
\(988\) −13.1851 −0.419473
\(989\) 8.37384 0.266273
\(990\) 0 0
\(991\) 2.46785 0.0783938 0.0391969 0.999232i \(-0.487520\pi\)
0.0391969 + 0.999232i \(0.487520\pi\)
\(992\) 22.2514 0.706483
\(993\) 0 0
\(994\) −0.256757 −0.00814383
\(995\) 67.8866 2.15215
\(996\) 0 0
\(997\) −57.3747 −1.81708 −0.908538 0.417803i \(-0.862800\pi\)
−0.908538 + 0.417803i \(0.862800\pi\)
\(998\) 1.60307 0.0507441
\(999\) 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 2151.2.a.j.1.10 20
3.2 odd 2 2151.2.a.k.1.11 yes 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.10 20 1.1 even 1 trivial
2151.2.a.k.1.11 yes 20 3.2 odd 2