Properties

Label 2151.2.a.e.1.2
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $1$
Dimension $6$
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: \(6\)
Coefficient field: 6.6.1767625.1
Defining polynomial: \(x^{6} - 7 x^{4} - x^{3} + 11 x^{2} + x - 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 717)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(2.30642\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.05161 q^{2} -0.894124 q^{4} -2.87285 q^{5} -4.11383 q^{7} +3.04348 q^{8} +O(q^{10})\) \(q-1.05161 q^{2} -0.894124 q^{4} -2.87285 q^{5} -4.11383 q^{7} +3.04348 q^{8} +3.02110 q^{10} +1.17944 q^{11} +0.346441 q^{13} +4.32613 q^{14} -1.41230 q^{16} +3.26277 q^{17} -1.31974 q^{19} +2.56868 q^{20} -1.24031 q^{22} +9.10667 q^{23} +3.25325 q^{25} -0.364319 q^{26} +3.67827 q^{28} +0.639311 q^{29} -1.52951 q^{31} -4.60178 q^{32} -3.43115 q^{34} +11.8184 q^{35} +1.40670 q^{37} +1.38785 q^{38} -8.74345 q^{40} +2.17524 q^{41} -1.97537 q^{43} -1.05456 q^{44} -9.57663 q^{46} +1.69565 q^{47} +9.92360 q^{49} -3.42114 q^{50} -0.309761 q^{52} +0.0493538 q^{53} -3.38835 q^{55} -12.5204 q^{56} -0.672304 q^{58} -0.806525 q^{59} -1.25891 q^{61} +1.60844 q^{62} +7.66385 q^{64} -0.995271 q^{65} +4.01099 q^{67} -2.91732 q^{68} -12.4283 q^{70} -0.361936 q^{71} -11.9650 q^{73} -1.47930 q^{74} +1.18001 q^{76} -4.85201 q^{77} +4.27659 q^{79} +4.05731 q^{80} -2.28749 q^{82} -14.0559 q^{83} -9.37343 q^{85} +2.07731 q^{86} +3.58960 q^{88} +10.6550 q^{89} -1.42520 q^{91} -8.14249 q^{92} -1.78316 q^{94} +3.79142 q^{95} -10.2134 q^{97} -10.4357 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} + O(q^{10}) \) \( 6q + 2q^{2} + 4q^{4} - 5q^{5} - 9q^{7} + 3q^{8} - 11q^{10} + 13q^{11} - q^{13} - 4q^{16} - 11q^{17} - 22q^{19} + q^{20} - 2q^{22} + 12q^{23} - q^{25} - 12q^{26} - 16q^{28} - 18q^{31} - 7q^{32} - 3q^{34} + 9q^{35} - 8q^{37} + 5q^{38} - 11q^{40} - 10q^{41} - 14q^{43} + 4q^{44} - 18q^{46} + 9q^{47} + 5q^{49} - 4q^{50} - 16q^{52} + 8q^{53} - 20q^{55} - 11q^{56} - 15q^{58} + 10q^{59} - 12q^{61} + 13q^{62} - 31q^{64} + 11q^{65} - 36q^{67} - 22q^{68} + q^{70} + 3q^{71} - 32q^{73} - 9q^{74} - 4q^{76} - 6q^{77} - q^{79} + 7q^{80} + 7q^{82} + 7q^{83} - 14q^{85} - 45q^{86} - 15q^{88} - 17q^{89} - 23q^{91} + 12q^{92} + 50q^{94} - 28q^{97} - 13q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.05161 −0.743598 −0.371799 0.928313i \(-0.621259\pi\)
−0.371799 + 0.928313i \(0.621259\pi\)
\(3\) 0 0
\(4\) −0.894124 −0.447062
\(5\) −2.87285 −1.28478 −0.642388 0.766380i \(-0.722056\pi\)
−0.642388 + 0.766380i \(0.722056\pi\)
\(6\) 0 0
\(7\) −4.11383 −1.55488 −0.777441 0.628956i \(-0.783483\pi\)
−0.777441 + 0.628956i \(0.783483\pi\)
\(8\) 3.04348 1.07603
\(9\) 0 0
\(10\) 3.02110 0.955357
\(11\) 1.17944 0.355614 0.177807 0.984065i \(-0.443100\pi\)
0.177807 + 0.984065i \(0.443100\pi\)
\(12\) 0 0
\(13\) 0.346441 0.0960854 0.0480427 0.998845i \(-0.484702\pi\)
0.0480427 + 0.998845i \(0.484702\pi\)
\(14\) 4.32613 1.15621
\(15\) 0 0
\(16\) −1.41230 −0.353074
\(17\) 3.26277 0.791337 0.395669 0.918393i \(-0.370513\pi\)
0.395669 + 0.918393i \(0.370513\pi\)
\(18\) 0 0
\(19\) −1.31974 −0.302770 −0.151385 0.988475i \(-0.548373\pi\)
−0.151385 + 0.988475i \(0.548373\pi\)
\(20\) 2.56868 0.574374
\(21\) 0 0
\(22\) −1.24031 −0.264434
\(23\) 9.10667 1.89887 0.949436 0.313960i \(-0.101656\pi\)
0.949436 + 0.313960i \(0.101656\pi\)
\(24\) 0 0
\(25\) 3.25325 0.650649
\(26\) −0.364319 −0.0714489
\(27\) 0 0
\(28\) 3.67827 0.695128
\(29\) 0.639311 0.118717 0.0593585 0.998237i \(-0.481094\pi\)
0.0593585 + 0.998237i \(0.481094\pi\)
\(30\) 0 0
\(31\) −1.52951 −0.274708 −0.137354 0.990522i \(-0.543860\pi\)
−0.137354 + 0.990522i \(0.543860\pi\)
\(32\) −4.60178 −0.813487
\(33\) 0 0
\(34\) −3.43115 −0.588437
\(35\) 11.8184 1.99767
\(36\) 0 0
\(37\) 1.40670 0.231260 0.115630 0.993292i \(-0.463111\pi\)
0.115630 + 0.993292i \(0.463111\pi\)
\(38\) 1.38785 0.225139
\(39\) 0 0
\(40\) −8.74345 −1.38246
\(41\) 2.17524 0.339715 0.169857 0.985469i \(-0.445669\pi\)
0.169857 + 0.985469i \(0.445669\pi\)
\(42\) 0 0
\(43\) −1.97537 −0.301241 −0.150621 0.988592i \(-0.548127\pi\)
−0.150621 + 0.988592i \(0.548127\pi\)
\(44\) −1.05456 −0.158982
\(45\) 0 0
\(46\) −9.57663 −1.41200
\(47\) 1.69565 0.247336 0.123668 0.992324i \(-0.460534\pi\)
0.123668 + 0.992324i \(0.460534\pi\)
\(48\) 0 0
\(49\) 9.92360 1.41766
\(50\) −3.42114 −0.483822
\(51\) 0 0
\(52\) −0.309761 −0.0429561
\(53\) 0.0493538 0.00677926 0.00338963 0.999994i \(-0.498921\pi\)
0.00338963 + 0.999994i \(0.498921\pi\)
\(54\) 0 0
\(55\) −3.38835 −0.456884
\(56\) −12.5204 −1.67310
\(57\) 0 0
\(58\) −0.672304 −0.0882778
\(59\) −0.806525 −0.105001 −0.0525003 0.998621i \(-0.516719\pi\)
−0.0525003 + 0.998621i \(0.516719\pi\)
\(60\) 0 0
\(61\) −1.25891 −0.161187 −0.0805935 0.996747i \(-0.525682\pi\)
−0.0805935 + 0.996747i \(0.525682\pi\)
\(62\) 1.60844 0.204272
\(63\) 0 0
\(64\) 7.66385 0.957982
\(65\) −0.995271 −0.123448
\(66\) 0 0
\(67\) 4.01099 0.490021 0.245010 0.969520i \(-0.421209\pi\)
0.245010 + 0.969520i \(0.421209\pi\)
\(68\) −2.91732 −0.353777
\(69\) 0 0
\(70\) −12.4283 −1.48547
\(71\) −0.361936 −0.0429539 −0.0214770 0.999769i \(-0.506837\pi\)
−0.0214770 + 0.999769i \(0.506837\pi\)
\(72\) 0 0
\(73\) −11.9650 −1.40040 −0.700198 0.713949i \(-0.746905\pi\)
−0.700198 + 0.713949i \(0.746905\pi\)
\(74\) −1.47930 −0.171965
\(75\) 0 0
\(76\) 1.18001 0.135357
\(77\) −4.85201 −0.552938
\(78\) 0 0
\(79\) 4.27659 0.481154 0.240577 0.970630i \(-0.422663\pi\)
0.240577 + 0.970630i \(0.422663\pi\)
\(80\) 4.05731 0.453621
\(81\) 0 0
\(82\) −2.28749 −0.252611
\(83\) −14.0559 −1.54284 −0.771418 0.636329i \(-0.780452\pi\)
−0.771418 + 0.636329i \(0.780452\pi\)
\(84\) 0 0
\(85\) −9.37343 −1.01669
\(86\) 2.07731 0.224003
\(87\) 0 0
\(88\) 3.58960 0.382652
\(89\) 10.6550 1.12943 0.564714 0.825287i \(-0.308987\pi\)
0.564714 + 0.825287i \(0.308987\pi\)
\(90\) 0 0
\(91\) −1.42520 −0.149401
\(92\) −8.14249 −0.848913
\(93\) 0 0
\(94\) −1.78316 −0.183919
\(95\) 3.79142 0.388991
\(96\) 0 0
\(97\) −10.2134 −1.03701 −0.518505 0.855075i \(-0.673511\pi\)
−0.518505 + 0.855075i \(0.673511\pi\)
\(98\) −10.4357 −1.05417
\(99\) 0 0
\(100\) −2.90881 −0.290881
\(101\) −13.3232 −1.32571 −0.662855 0.748748i \(-0.730655\pi\)
−0.662855 + 0.748748i \(0.730655\pi\)
\(102\) 0 0
\(103\) −16.3775 −1.61372 −0.806860 0.590743i \(-0.798835\pi\)
−0.806860 + 0.590743i \(0.798835\pi\)
\(104\) 1.05439 0.103391
\(105\) 0 0
\(106\) −0.0519008 −0.00504105
\(107\) 10.8242 1.04641 0.523205 0.852207i \(-0.324736\pi\)
0.523205 + 0.852207i \(0.324736\pi\)
\(108\) 0 0
\(109\) 17.1432 1.64202 0.821012 0.570910i \(-0.193410\pi\)
0.821012 + 0.570910i \(0.193410\pi\)
\(110\) 3.56321 0.339738
\(111\) 0 0
\(112\) 5.80994 0.548988
\(113\) −13.1922 −1.24102 −0.620511 0.784198i \(-0.713075\pi\)
−0.620511 + 0.784198i \(0.713075\pi\)
\(114\) 0 0
\(115\) −26.1621 −2.43963
\(116\) −0.571623 −0.0530739
\(117\) 0 0
\(118\) 0.848147 0.0780782
\(119\) −13.4225 −1.23044
\(120\) 0 0
\(121\) −9.60892 −0.873539
\(122\) 1.32388 0.119858
\(123\) 0 0
\(124\) 1.36757 0.122812
\(125\) 5.01815 0.448837
\(126\) 0 0
\(127\) 13.0405 1.15716 0.578580 0.815625i \(-0.303607\pi\)
0.578580 + 0.815625i \(0.303607\pi\)
\(128\) 1.14420 0.101134
\(129\) 0 0
\(130\) 1.04663 0.0917959
\(131\) −20.5593 −1.79628 −0.898138 0.439714i \(-0.855080\pi\)
−0.898138 + 0.439714i \(0.855080\pi\)
\(132\) 0 0
\(133\) 5.42920 0.470771
\(134\) −4.21799 −0.364378
\(135\) 0 0
\(136\) 9.93016 0.851504
\(137\) −20.2062 −1.72633 −0.863165 0.504923i \(-0.831521\pi\)
−0.863165 + 0.504923i \(0.831521\pi\)
\(138\) 0 0
\(139\) 14.9894 1.27139 0.635694 0.771941i \(-0.280714\pi\)
0.635694 + 0.771941i \(0.280714\pi\)
\(140\) −10.5671 −0.893084
\(141\) 0 0
\(142\) 0.380614 0.0319404
\(143\) 0.408606 0.0341693
\(144\) 0 0
\(145\) −1.83664 −0.152525
\(146\) 12.5825 1.04133
\(147\) 0 0
\(148\) −1.25776 −0.103388
\(149\) −5.24611 −0.429778 −0.214889 0.976638i \(-0.568939\pi\)
−0.214889 + 0.976638i \(0.568939\pi\)
\(150\) 0 0
\(151\) 5.59004 0.454911 0.227456 0.973788i \(-0.426959\pi\)
0.227456 + 0.973788i \(0.426959\pi\)
\(152\) −4.01661 −0.325790
\(153\) 0 0
\(154\) 5.10241 0.411164
\(155\) 4.39405 0.352938
\(156\) 0 0
\(157\) 17.6356 1.40747 0.703736 0.710461i \(-0.251514\pi\)
0.703736 + 0.710461i \(0.251514\pi\)
\(158\) −4.49729 −0.357785
\(159\) 0 0
\(160\) 13.2202 1.04515
\(161\) −37.4633 −2.95252
\(162\) 0 0
\(163\) −12.8967 −1.01015 −0.505073 0.863077i \(-0.668534\pi\)
−0.505073 + 0.863077i \(0.668534\pi\)
\(164\) −1.94493 −0.151874
\(165\) 0 0
\(166\) 14.7813 1.14725
\(167\) 4.91384 0.380244 0.190122 0.981760i \(-0.439112\pi\)
0.190122 + 0.981760i \(0.439112\pi\)
\(168\) 0 0
\(169\) −12.8800 −0.990768
\(170\) 9.85716 0.756010
\(171\) 0 0
\(172\) 1.76623 0.134674
\(173\) 15.5398 1.18147 0.590736 0.806865i \(-0.298837\pi\)
0.590736 + 0.806865i \(0.298837\pi\)
\(174\) 0 0
\(175\) −13.3833 −1.01168
\(176\) −1.66572 −0.125558
\(177\) 0 0
\(178\) −11.2049 −0.839840
\(179\) 16.9641 1.26796 0.633979 0.773350i \(-0.281420\pi\)
0.633979 + 0.773350i \(0.281420\pi\)
\(180\) 0 0
\(181\) 3.30222 0.245452 0.122726 0.992441i \(-0.460836\pi\)
0.122726 + 0.992441i \(0.460836\pi\)
\(182\) 1.49875 0.111095
\(183\) 0 0
\(184\) 27.7160 2.04325
\(185\) −4.04124 −0.297118
\(186\) 0 0
\(187\) 3.84823 0.281411
\(188\) −1.51612 −0.110575
\(189\) 0 0
\(190\) −3.98708 −0.289253
\(191\) 12.2407 0.885704 0.442852 0.896595i \(-0.353967\pi\)
0.442852 + 0.896595i \(0.353967\pi\)
\(192\) 0 0
\(193\) −15.3502 −1.10493 −0.552465 0.833536i \(-0.686313\pi\)
−0.552465 + 0.833536i \(0.686313\pi\)
\(194\) 10.7404 0.771119
\(195\) 0 0
\(196\) −8.87293 −0.633781
\(197\) 20.6755 1.47307 0.736533 0.676402i \(-0.236462\pi\)
0.736533 + 0.676402i \(0.236462\pi\)
\(198\) 0 0
\(199\) −24.9536 −1.76892 −0.884459 0.466619i \(-0.845472\pi\)
−0.884459 + 0.466619i \(0.845472\pi\)
\(200\) 9.90119 0.700120
\(201\) 0 0
\(202\) 14.0108 0.985795
\(203\) −2.63002 −0.184591
\(204\) 0 0
\(205\) −6.24912 −0.436457
\(206\) 17.2226 1.19996
\(207\) 0 0
\(208\) −0.489277 −0.0339252
\(209\) −1.55655 −0.107669
\(210\) 0 0
\(211\) −3.86876 −0.266336 −0.133168 0.991093i \(-0.542515\pi\)
−0.133168 + 0.991093i \(0.542515\pi\)
\(212\) −0.0441284 −0.00303075
\(213\) 0 0
\(214\) −11.3828 −0.778109
\(215\) 5.67494 0.387028
\(216\) 0 0
\(217\) 6.29214 0.427139
\(218\) −18.0279 −1.22101
\(219\) 0 0
\(220\) 3.02960 0.204256
\(221\) 1.13036 0.0760360
\(222\) 0 0
\(223\) 9.50369 0.636414 0.318207 0.948021i \(-0.396919\pi\)
0.318207 + 0.948021i \(0.396919\pi\)
\(224\) 18.9309 1.26488
\(225\) 0 0
\(226\) 13.8730 0.922821
\(227\) −7.95147 −0.527758 −0.263879 0.964556i \(-0.585002\pi\)
−0.263879 + 0.964556i \(0.585002\pi\)
\(228\) 0 0
\(229\) −11.1993 −0.740070 −0.370035 0.929018i \(-0.620654\pi\)
−0.370035 + 0.929018i \(0.620654\pi\)
\(230\) 27.5122 1.81410
\(231\) 0 0
\(232\) 1.94573 0.127743
\(233\) 15.6891 1.02783 0.513914 0.857842i \(-0.328195\pi\)
0.513914 + 0.857842i \(0.328195\pi\)
\(234\) 0 0
\(235\) −4.87135 −0.317772
\(236\) 0.721133 0.0469418
\(237\) 0 0
\(238\) 14.1152 0.914950
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −25.5643 −1.64674 −0.823371 0.567503i \(-0.807910\pi\)
−0.823371 + 0.567503i \(0.807910\pi\)
\(242\) 10.1048 0.649562
\(243\) 0 0
\(244\) 1.12562 0.0720605
\(245\) −28.5090 −1.82137
\(246\) 0 0
\(247\) −0.457213 −0.0290917
\(248\) −4.65503 −0.295595
\(249\) 0 0
\(250\) −5.27712 −0.333754
\(251\) −16.7833 −1.05935 −0.529676 0.848200i \(-0.677686\pi\)
−0.529676 + 0.848200i \(0.677686\pi\)
\(252\) 0 0
\(253\) 10.7408 0.675266
\(254\) −13.7135 −0.860462
\(255\) 0 0
\(256\) −16.5310 −1.03318
\(257\) 6.98956 0.435996 0.217998 0.975949i \(-0.430047\pi\)
0.217998 + 0.975949i \(0.430047\pi\)
\(258\) 0 0
\(259\) −5.78693 −0.359582
\(260\) 0.889896 0.0551890
\(261\) 0 0
\(262\) 21.6203 1.33571
\(263\) 13.7752 0.849413 0.424707 0.905331i \(-0.360377\pi\)
0.424707 + 0.905331i \(0.360377\pi\)
\(264\) 0 0
\(265\) −0.141786 −0.00870984
\(266\) −5.70938 −0.350064
\(267\) 0 0
\(268\) −3.58632 −0.219070
\(269\) −11.2058 −0.683228 −0.341614 0.939840i \(-0.610974\pi\)
−0.341614 + 0.939840i \(0.610974\pi\)
\(270\) 0 0
\(271\) −20.5528 −1.24850 −0.624248 0.781226i \(-0.714595\pi\)
−0.624248 + 0.781226i \(0.714595\pi\)
\(272\) −4.60799 −0.279400
\(273\) 0 0
\(274\) 21.2489 1.28370
\(275\) 3.83701 0.231380
\(276\) 0 0
\(277\) −17.2928 −1.03902 −0.519511 0.854463i \(-0.673886\pi\)
−0.519511 + 0.854463i \(0.673886\pi\)
\(278\) −15.7630 −0.945401
\(279\) 0 0
\(280\) 35.9691 2.14956
\(281\) −24.1530 −1.44085 −0.720425 0.693533i \(-0.756053\pi\)
−0.720425 + 0.693533i \(0.756053\pi\)
\(282\) 0 0
\(283\) −12.4777 −0.741722 −0.370861 0.928688i \(-0.620937\pi\)
−0.370861 + 0.928688i \(0.620937\pi\)
\(284\) 0.323616 0.0192031
\(285\) 0 0
\(286\) −0.429692 −0.0254082
\(287\) −8.94855 −0.528216
\(288\) 0 0
\(289\) −6.35435 −0.373785
\(290\) 1.93143 0.113417
\(291\) 0 0
\(292\) 10.6982 0.626063
\(293\) 24.0095 1.40265 0.701325 0.712842i \(-0.252592\pi\)
0.701325 + 0.712842i \(0.252592\pi\)
\(294\) 0 0
\(295\) 2.31702 0.134902
\(296\) 4.28126 0.248843
\(297\) 0 0
\(298\) 5.51685 0.319582
\(299\) 3.15492 0.182454
\(300\) 0 0
\(301\) 8.12635 0.468395
\(302\) −5.87853 −0.338271
\(303\) 0 0
\(304\) 1.86387 0.106900
\(305\) 3.61666 0.207089
\(306\) 0 0
\(307\) 7.54396 0.430557 0.215278 0.976553i \(-0.430934\pi\)
0.215278 + 0.976553i \(0.430934\pi\)
\(308\) 4.33830 0.247197
\(309\) 0 0
\(310\) −4.62081 −0.262444
\(311\) 7.22920 0.409930 0.204965 0.978769i \(-0.434292\pi\)
0.204965 + 0.978769i \(0.434292\pi\)
\(312\) 0 0
\(313\) −21.2712 −1.20232 −0.601161 0.799128i \(-0.705295\pi\)
−0.601161 + 0.799128i \(0.705295\pi\)
\(314\) −18.5457 −1.04659
\(315\) 0 0
\(316\) −3.82380 −0.215106
\(317\) −31.0998 −1.74674 −0.873369 0.487059i \(-0.838070\pi\)
−0.873369 + 0.487059i \(0.838070\pi\)
\(318\) 0 0
\(319\) 0.754028 0.0422175
\(320\) −22.0171 −1.23079
\(321\) 0 0
\(322\) 39.3967 2.19549
\(323\) −4.30601 −0.239593
\(324\) 0 0
\(325\) 1.12706 0.0625179
\(326\) 13.5622 0.751142
\(327\) 0 0
\(328\) 6.62029 0.365544
\(329\) −6.97562 −0.384579
\(330\) 0 0
\(331\) −4.80463 −0.264086 −0.132043 0.991244i \(-0.542154\pi\)
−0.132043 + 0.991244i \(0.542154\pi\)
\(332\) 12.5677 0.689743
\(333\) 0 0
\(334\) −5.16743 −0.282749
\(335\) −11.5230 −0.629567
\(336\) 0 0
\(337\) −1.20117 −0.0654317 −0.0327159 0.999465i \(-0.510416\pi\)
−0.0327159 + 0.999465i \(0.510416\pi\)
\(338\) 13.5447 0.736733
\(339\) 0 0
\(340\) 8.38100 0.454524
\(341\) −1.80396 −0.0976901
\(342\) 0 0
\(343\) −12.0272 −0.649408
\(344\) −6.01200 −0.324146
\(345\) 0 0
\(346\) −16.3418 −0.878541
\(347\) 1.61746 0.0868296 0.0434148 0.999057i \(-0.486176\pi\)
0.0434148 + 0.999057i \(0.486176\pi\)
\(348\) 0 0
\(349\) 31.2253 1.67145 0.835727 0.549145i \(-0.185047\pi\)
0.835727 + 0.549145i \(0.185047\pi\)
\(350\) 14.0740 0.752286
\(351\) 0 0
\(352\) −5.42752 −0.289288
\(353\) 14.6883 0.781779 0.390889 0.920438i \(-0.372168\pi\)
0.390889 + 0.920438i \(0.372168\pi\)
\(354\) 0 0
\(355\) 1.03979 0.0551861
\(356\) −9.52689 −0.504924
\(357\) 0 0
\(358\) −17.8396 −0.942851
\(359\) −6.81390 −0.359624 −0.179812 0.983701i \(-0.557549\pi\)
−0.179812 + 0.983701i \(0.557549\pi\)
\(360\) 0 0
\(361\) −17.2583 −0.908331
\(362\) −3.47264 −0.182518
\(363\) 0 0
\(364\) 1.27430 0.0667917
\(365\) 34.3736 1.79919
\(366\) 0 0
\(367\) 9.51165 0.496504 0.248252 0.968695i \(-0.420144\pi\)
0.248252 + 0.968695i \(0.420144\pi\)
\(368\) −12.8613 −0.670442
\(369\) 0 0
\(370\) 4.24979 0.220936
\(371\) −0.203033 −0.0105410
\(372\) 0 0
\(373\) −12.7503 −0.660186 −0.330093 0.943949i \(-0.607080\pi\)
−0.330093 + 0.943949i \(0.607080\pi\)
\(374\) −4.04683 −0.209256
\(375\) 0 0
\(376\) 5.16068 0.266142
\(377\) 0.221483 0.0114070
\(378\) 0 0
\(379\) −10.2372 −0.525852 −0.262926 0.964816i \(-0.584688\pi\)
−0.262926 + 0.964816i \(0.584688\pi\)
\(380\) −3.39000 −0.173903
\(381\) 0 0
\(382\) −12.8724 −0.658608
\(383\) 24.8455 1.26954 0.634772 0.772699i \(-0.281094\pi\)
0.634772 + 0.772699i \(0.281094\pi\)
\(384\) 0 0
\(385\) 13.9391 0.710401
\(386\) 16.1423 0.821624
\(387\) 0 0
\(388\) 9.13201 0.463608
\(389\) −3.01669 −0.152952 −0.0764760 0.997071i \(-0.524367\pi\)
−0.0764760 + 0.997071i \(0.524367\pi\)
\(390\) 0 0
\(391\) 29.7129 1.50265
\(392\) 30.2023 1.52545
\(393\) 0 0
\(394\) −21.7424 −1.09537
\(395\) −12.2860 −0.618175
\(396\) 0 0
\(397\) 0.428312 0.0214964 0.0107482 0.999942i \(-0.496579\pi\)
0.0107482 + 0.999942i \(0.496579\pi\)
\(398\) 26.2414 1.31536
\(399\) 0 0
\(400\) −4.59455 −0.229727
\(401\) 14.0036 0.699305 0.349653 0.936879i \(-0.386300\pi\)
0.349653 + 0.936879i \(0.386300\pi\)
\(402\) 0 0
\(403\) −0.529885 −0.0263954
\(404\) 11.9126 0.592674
\(405\) 0 0
\(406\) 2.76574 0.137262
\(407\) 1.65912 0.0822394
\(408\) 0 0
\(409\) −39.9092 −1.97338 −0.986692 0.162600i \(-0.948012\pi\)
−0.986692 + 0.162600i \(0.948012\pi\)
\(410\) 6.57162 0.324549
\(411\) 0 0
\(412\) 14.6435 0.721432
\(413\) 3.31791 0.163264
\(414\) 0 0
\(415\) 40.3805 1.98220
\(416\) −1.59424 −0.0781643
\(417\) 0 0
\(418\) 1.63688 0.0800626
\(419\) −23.6201 −1.15392 −0.576959 0.816773i \(-0.695761\pi\)
−0.576959 + 0.816773i \(0.695761\pi\)
\(420\) 0 0
\(421\) −16.4261 −0.800559 −0.400279 0.916393i \(-0.631087\pi\)
−0.400279 + 0.916393i \(0.631087\pi\)
\(422\) 4.06841 0.198047
\(423\) 0 0
\(424\) 0.150207 0.00729471
\(425\) 10.6146 0.514883
\(426\) 0 0
\(427\) 5.17894 0.250627
\(428\) −9.67814 −0.467810
\(429\) 0 0
\(430\) −5.96780 −0.287793
\(431\) −24.1771 −1.16457 −0.582285 0.812985i \(-0.697841\pi\)
−0.582285 + 0.812985i \(0.697841\pi\)
\(432\) 0 0
\(433\) −7.20673 −0.346333 −0.173167 0.984893i \(-0.555400\pi\)
−0.173167 + 0.984893i \(0.555400\pi\)
\(434\) −6.61686 −0.317619
\(435\) 0 0
\(436\) −15.3282 −0.734087
\(437\) −12.0185 −0.574921
\(438\) 0 0
\(439\) 21.1400 1.00896 0.504479 0.863424i \(-0.331684\pi\)
0.504479 + 0.863424i \(0.331684\pi\)
\(440\) −10.3124 −0.491623
\(441\) 0 0
\(442\) −1.18869 −0.0565402
\(443\) 11.1509 0.529797 0.264899 0.964276i \(-0.414662\pi\)
0.264899 + 0.964276i \(0.414662\pi\)
\(444\) 0 0
\(445\) −30.6102 −1.45106
\(446\) −9.99415 −0.473237
\(447\) 0 0
\(448\) −31.5278 −1.48955
\(449\) 2.12495 0.100283 0.0501414 0.998742i \(-0.484033\pi\)
0.0501414 + 0.998742i \(0.484033\pi\)
\(450\) 0 0
\(451\) 2.56556 0.120807
\(452\) 11.7955 0.554814
\(453\) 0 0
\(454\) 8.36182 0.392440
\(455\) 4.09438 0.191947
\(456\) 0 0
\(457\) 37.1322 1.73697 0.868485 0.495715i \(-0.165094\pi\)
0.868485 + 0.495715i \(0.165094\pi\)
\(458\) 11.7772 0.550315
\(459\) 0 0
\(460\) 23.3921 1.09066
\(461\) −29.8834 −1.39181 −0.695905 0.718134i \(-0.744996\pi\)
−0.695905 + 0.718134i \(0.744996\pi\)
\(462\) 0 0
\(463\) 20.4052 0.948311 0.474156 0.880441i \(-0.342753\pi\)
0.474156 + 0.880441i \(0.342753\pi\)
\(464\) −0.902896 −0.0419159
\(465\) 0 0
\(466\) −16.4988 −0.764291
\(467\) −4.72528 −0.218660 −0.109330 0.994006i \(-0.534871\pi\)
−0.109330 + 0.994006i \(0.534871\pi\)
\(468\) 0 0
\(469\) −16.5005 −0.761924
\(470\) 5.12274 0.236294
\(471\) 0 0
\(472\) −2.45464 −0.112984
\(473\) −2.32983 −0.107126
\(474\) 0 0
\(475\) −4.29345 −0.196997
\(476\) 12.0013 0.550081
\(477\) 0 0
\(478\) −1.05161 −0.0480994
\(479\) −39.4927 −1.80447 −0.902234 0.431247i \(-0.858074\pi\)
−0.902234 + 0.431247i \(0.858074\pi\)
\(480\) 0 0
\(481\) 0.487339 0.0222207
\(482\) 26.8836 1.22451
\(483\) 0 0
\(484\) 8.59157 0.390526
\(485\) 29.3414 1.33233
\(486\) 0 0
\(487\) −26.1172 −1.18349 −0.591743 0.806127i \(-0.701560\pi\)
−0.591743 + 0.806127i \(0.701560\pi\)
\(488\) −3.83147 −0.173442
\(489\) 0 0
\(490\) 29.9802 1.35437
\(491\) 20.1522 0.909454 0.454727 0.890631i \(-0.349737\pi\)
0.454727 + 0.890631i \(0.349737\pi\)
\(492\) 0 0
\(493\) 2.08592 0.0939453
\(494\) 0.480808 0.0216326
\(495\) 0 0
\(496\) 2.16012 0.0969922
\(497\) 1.48894 0.0667882
\(498\) 0 0
\(499\) 16.6298 0.744452 0.372226 0.928142i \(-0.378595\pi\)
0.372226 + 0.928142i \(0.378595\pi\)
\(500\) −4.48685 −0.200658
\(501\) 0 0
\(502\) 17.6494 0.787732
\(503\) 13.0732 0.582906 0.291453 0.956585i \(-0.405861\pi\)
0.291453 + 0.956585i \(0.405861\pi\)
\(504\) 0 0
\(505\) 38.2755 1.70324
\(506\) −11.2951 −0.502126
\(507\) 0 0
\(508\) −11.6599 −0.517322
\(509\) 12.4973 0.553934 0.276967 0.960879i \(-0.410671\pi\)
0.276967 + 0.960879i \(0.410671\pi\)
\(510\) 0 0
\(511\) 49.2219 2.17745
\(512\) 15.0957 0.667140
\(513\) 0 0
\(514\) −7.35026 −0.324206
\(515\) 47.0499 2.07327
\(516\) 0 0
\(517\) 1.99992 0.0879563
\(518\) 6.08557 0.267385
\(519\) 0 0
\(520\) −3.02909 −0.132834
\(521\) −3.31677 −0.145310 −0.0726552 0.997357i \(-0.523147\pi\)
−0.0726552 + 0.997357i \(0.523147\pi\)
\(522\) 0 0
\(523\) 32.7979 1.43415 0.717075 0.696996i \(-0.245480\pi\)
0.717075 + 0.696996i \(0.245480\pi\)
\(524\) 18.3826 0.803047
\(525\) 0 0
\(526\) −14.4861 −0.631622
\(527\) −4.99043 −0.217387
\(528\) 0 0
\(529\) 59.9315 2.60572
\(530\) 0.149103 0.00647662
\(531\) 0 0
\(532\) −4.85437 −0.210464
\(533\) 0.753591 0.0326416
\(534\) 0 0
\(535\) −31.0961 −1.34440
\(536\) 12.2074 0.527278
\(537\) 0 0
\(538\) 11.7841 0.508047
\(539\) 11.7043 0.504139
\(540\) 0 0
\(541\) −5.52107 −0.237369 −0.118685 0.992932i \(-0.537868\pi\)
−0.118685 + 0.992932i \(0.537868\pi\)
\(542\) 21.6135 0.928380
\(543\) 0 0
\(544\) −15.0145 −0.643743
\(545\) −49.2499 −2.10963
\(546\) 0 0
\(547\) 12.7057 0.543258 0.271629 0.962402i \(-0.412438\pi\)
0.271629 + 0.962402i \(0.412438\pi\)
\(548\) 18.0668 0.771776
\(549\) 0 0
\(550\) −4.03502 −0.172054
\(551\) −0.843726 −0.0359439
\(552\) 0 0
\(553\) −17.5932 −0.748138
\(554\) 18.1852 0.772615
\(555\) 0 0
\(556\) −13.4024 −0.568389
\(557\) −42.9858 −1.82137 −0.910683 0.413105i \(-0.864444\pi\)
−0.910683 + 0.413105i \(0.864444\pi\)
\(558\) 0 0
\(559\) −0.684350 −0.0289449
\(560\) −16.6911 −0.705327
\(561\) 0 0
\(562\) 25.3995 1.07141
\(563\) −20.3703 −0.858504 −0.429252 0.903185i \(-0.641223\pi\)
−0.429252 + 0.903185i \(0.641223\pi\)
\(564\) 0 0
\(565\) 37.8993 1.59444
\(566\) 13.1216 0.551543
\(567\) 0 0
\(568\) −1.10154 −0.0462198
\(569\) 31.4900 1.32013 0.660064 0.751210i \(-0.270529\pi\)
0.660064 + 0.751210i \(0.270529\pi\)
\(570\) 0 0
\(571\) −31.3410 −1.31158 −0.655789 0.754944i \(-0.727664\pi\)
−0.655789 + 0.754944i \(0.727664\pi\)
\(572\) −0.365344 −0.0152758
\(573\) 0 0
\(574\) 9.41036 0.392781
\(575\) 29.6263 1.23550
\(576\) 0 0
\(577\) 23.8525 0.992992 0.496496 0.868039i \(-0.334620\pi\)
0.496496 + 0.868039i \(0.334620\pi\)
\(578\) 6.68228 0.277946
\(579\) 0 0
\(580\) 1.64219 0.0681881
\(581\) 57.8236 2.39893
\(582\) 0 0
\(583\) 0.0582098 0.00241080
\(584\) −36.4152 −1.50687
\(585\) 0 0
\(586\) −25.2485 −1.04301
\(587\) −8.14382 −0.336131 −0.168066 0.985776i \(-0.553752\pi\)
−0.168066 + 0.985776i \(0.553752\pi\)
\(588\) 0 0
\(589\) 2.01856 0.0831733
\(590\) −2.43660 −0.100313
\(591\) 0 0
\(592\) −1.98668 −0.0816519
\(593\) −1.77048 −0.0727049 −0.0363524 0.999339i \(-0.511574\pi\)
−0.0363524 + 0.999339i \(0.511574\pi\)
\(594\) 0 0
\(595\) 38.5607 1.58083
\(596\) 4.69067 0.192138
\(597\) 0 0
\(598\) −3.31774 −0.135672
\(599\) 24.1665 0.987415 0.493707 0.869628i \(-0.335641\pi\)
0.493707 + 0.869628i \(0.335641\pi\)
\(600\) 0 0
\(601\) 10.2259 0.417123 0.208561 0.978009i \(-0.433122\pi\)
0.208561 + 0.978009i \(0.433122\pi\)
\(602\) −8.54572 −0.348297
\(603\) 0 0
\(604\) −4.99819 −0.203373
\(605\) 27.6050 1.12230
\(606\) 0 0
\(607\) −21.0640 −0.854960 −0.427480 0.904025i \(-0.640599\pi\)
−0.427480 + 0.904025i \(0.640599\pi\)
\(608\) 6.07316 0.246299
\(609\) 0 0
\(610\) −3.80330 −0.153991
\(611\) 0.587443 0.0237654
\(612\) 0 0
\(613\) −6.72545 −0.271638 −0.135819 0.990734i \(-0.543367\pi\)
−0.135819 + 0.990734i \(0.543367\pi\)
\(614\) −7.93328 −0.320161
\(615\) 0 0
\(616\) −14.7670 −0.594979
\(617\) 0.401847 0.0161778 0.00808888 0.999967i \(-0.497425\pi\)
0.00808888 + 0.999967i \(0.497425\pi\)
\(618\) 0 0
\(619\) −20.6279 −0.829104 −0.414552 0.910026i \(-0.636062\pi\)
−0.414552 + 0.910026i \(0.636062\pi\)
\(620\) −3.92882 −0.157785
\(621\) 0 0
\(622\) −7.60227 −0.304823
\(623\) −43.8329 −1.75613
\(624\) 0 0
\(625\) −30.6826 −1.22730
\(626\) 22.3690 0.894044
\(627\) 0 0
\(628\) −15.7684 −0.629227
\(629\) 4.58974 0.183005
\(630\) 0 0
\(631\) −29.6371 −1.17984 −0.589918 0.807463i \(-0.700840\pi\)
−0.589918 + 0.807463i \(0.700840\pi\)
\(632\) 13.0157 0.517737
\(633\) 0 0
\(634\) 32.7047 1.29887
\(635\) −37.4635 −1.48669
\(636\) 0 0
\(637\) 3.43794 0.136216
\(638\) −0.792941 −0.0313928
\(639\) 0 0
\(640\) −3.28712 −0.129935
\(641\) −11.3866 −0.449745 −0.224872 0.974388i \(-0.572197\pi\)
−0.224872 + 0.974388i \(0.572197\pi\)
\(642\) 0 0
\(643\) −33.4917 −1.32079 −0.660393 0.750921i \(-0.729610\pi\)
−0.660393 + 0.750921i \(0.729610\pi\)
\(644\) 33.4968 1.31996
\(645\) 0 0
\(646\) 4.52823 0.178161
\(647\) −48.9050 −1.92265 −0.961326 0.275412i \(-0.911186\pi\)
−0.961326 + 0.275412i \(0.911186\pi\)
\(648\) 0 0
\(649\) −0.951247 −0.0373397
\(650\) −1.18522 −0.0464882
\(651\) 0 0
\(652\) 11.5312 0.451598
\(653\) −5.23254 −0.204765 −0.102383 0.994745i \(-0.532647\pi\)
−0.102383 + 0.994745i \(0.532647\pi\)
\(654\) 0 0
\(655\) 59.0638 2.30781
\(656\) −3.07208 −0.119944
\(657\) 0 0
\(658\) 7.33561 0.285972
\(659\) 3.59290 0.139960 0.0699798 0.997548i \(-0.477707\pi\)
0.0699798 + 0.997548i \(0.477707\pi\)
\(660\) 0 0
\(661\) −15.9161 −0.619066 −0.309533 0.950889i \(-0.600173\pi\)
−0.309533 + 0.950889i \(0.600173\pi\)
\(662\) 5.05258 0.196374
\(663\) 0 0
\(664\) −42.7789 −1.66014
\(665\) −15.5972 −0.604835
\(666\) 0 0
\(667\) 5.82200 0.225429
\(668\) −4.39358 −0.169993
\(669\) 0 0
\(670\) 12.1176 0.468145
\(671\) −1.48481 −0.0573204
\(672\) 0 0
\(673\) 34.3829 1.32536 0.662682 0.748901i \(-0.269418\pi\)
0.662682 + 0.748901i \(0.269418\pi\)
\(674\) 1.26315 0.0486549
\(675\) 0 0
\(676\) 11.5163 0.442934
\(677\) 27.8827 1.07162 0.535809 0.844339i \(-0.320007\pi\)
0.535809 + 0.844339i \(0.320007\pi\)
\(678\) 0 0
\(679\) 42.0161 1.61243
\(680\) −28.5278 −1.09399
\(681\) 0 0
\(682\) 1.89706 0.0726422
\(683\) −1.84211 −0.0704864 −0.0352432 0.999379i \(-0.511221\pi\)
−0.0352432 + 0.999379i \(0.511221\pi\)
\(684\) 0 0
\(685\) 58.0492 2.21795
\(686\) 12.6479 0.482898
\(687\) 0 0
\(688\) 2.78981 0.106360
\(689\) 0.0170982 0.000651388 0
\(690\) 0 0
\(691\) 18.0372 0.686166 0.343083 0.939305i \(-0.388529\pi\)
0.343083 + 0.939305i \(0.388529\pi\)
\(692\) −13.8945 −0.528191
\(693\) 0 0
\(694\) −1.70093 −0.0645663
\(695\) −43.0623 −1.63345
\(696\) 0 0
\(697\) 7.09729 0.268829
\(698\) −32.8368 −1.24289
\(699\) 0 0
\(700\) 11.9663 0.452285
\(701\) 6.10724 0.230667 0.115334 0.993327i \(-0.463206\pi\)
0.115334 + 0.993327i \(0.463206\pi\)
\(702\) 0 0
\(703\) −1.85648 −0.0700186
\(704\) 9.03904 0.340672
\(705\) 0 0
\(706\) −15.4463 −0.581329
\(707\) 54.8094 2.06132
\(708\) 0 0
\(709\) −29.3897 −1.10375 −0.551877 0.833926i \(-0.686088\pi\)
−0.551877 + 0.833926i \(0.686088\pi\)
\(710\) −1.09345 −0.0410363
\(711\) 0 0
\(712\) 32.4283 1.21530
\(713\) −13.9287 −0.521636
\(714\) 0 0
\(715\) −1.17386 −0.0438999
\(716\) −15.1680 −0.566856
\(717\) 0 0
\(718\) 7.16554 0.267416
\(719\) 38.2319 1.42581 0.712904 0.701262i \(-0.247379\pi\)
0.712904 + 0.701262i \(0.247379\pi\)
\(720\) 0 0
\(721\) 67.3741 2.50914
\(722\) 18.1489 0.675433
\(723\) 0 0
\(724\) −2.95260 −0.109732
\(725\) 2.07984 0.0772432
\(726\) 0 0
\(727\) 16.8660 0.625526 0.312763 0.949831i \(-0.398745\pi\)
0.312763 + 0.949831i \(0.398745\pi\)
\(728\) −4.33756 −0.160761
\(729\) 0 0
\(730\) −36.1475 −1.33788
\(731\) −6.44518 −0.238384
\(732\) 0 0
\(733\) −9.13669 −0.337471 −0.168736 0.985661i \(-0.553968\pi\)
−0.168736 + 0.985661i \(0.553968\pi\)
\(734\) −10.0025 −0.369200
\(735\) 0 0
\(736\) −41.9069 −1.54471
\(737\) 4.73072 0.174258
\(738\) 0 0
\(739\) −28.4826 −1.04775 −0.523875 0.851795i \(-0.675514\pi\)
−0.523875 + 0.851795i \(0.675514\pi\)
\(740\) 3.61336 0.132830
\(741\) 0 0
\(742\) 0.213511 0.00783823
\(743\) 37.7287 1.38413 0.692066 0.721834i \(-0.256701\pi\)
0.692066 + 0.721834i \(0.256701\pi\)
\(744\) 0 0
\(745\) 15.0713 0.552169
\(746\) 13.4083 0.490913
\(747\) 0 0
\(748\) −3.44080 −0.125808
\(749\) −44.5288 −1.62705
\(750\) 0 0
\(751\) −32.0210 −1.16846 −0.584231 0.811588i \(-0.698604\pi\)
−0.584231 + 0.811588i \(0.698604\pi\)
\(752\) −2.39476 −0.0873280
\(753\) 0 0
\(754\) −0.232913 −0.00848221
\(755\) −16.0593 −0.584459
\(756\) 0 0
\(757\) −13.7562 −0.499976 −0.249988 0.968249i \(-0.580427\pi\)
−0.249988 + 0.968249i \(0.580427\pi\)
\(758\) 10.7656 0.391023
\(759\) 0 0
\(760\) 11.5391 0.418567
\(761\) −45.3489 −1.64390 −0.821948 0.569563i \(-0.807112\pi\)
−0.821948 + 0.569563i \(0.807112\pi\)
\(762\) 0 0
\(763\) −70.5244 −2.55315
\(764\) −10.9447 −0.395965
\(765\) 0 0
\(766\) −26.1277 −0.944031
\(767\) −0.279413 −0.0100890
\(768\) 0 0
\(769\) −15.6445 −0.564153 −0.282077 0.959392i \(-0.591023\pi\)
−0.282077 + 0.959392i \(0.591023\pi\)
\(770\) −14.6584 −0.528253
\(771\) 0 0
\(772\) 13.7250 0.493972
\(773\) 22.9648 0.825988 0.412994 0.910734i \(-0.364483\pi\)
0.412994 + 0.910734i \(0.364483\pi\)
\(774\) 0 0
\(775\) −4.97587 −0.178739
\(776\) −31.0842 −1.11586
\(777\) 0 0
\(778\) 3.17237 0.113735
\(779\) −2.87075 −0.102855
\(780\) 0 0
\(781\) −0.426881 −0.0152750
\(782\) −31.2463 −1.11737
\(783\) 0 0
\(784\) −14.0151 −0.500538
\(785\) −50.6643 −1.80829
\(786\) 0 0
\(787\) 11.3151 0.403341 0.201671 0.979453i \(-0.435363\pi\)
0.201671 + 0.979453i \(0.435363\pi\)
\(788\) −18.4864 −0.658551
\(789\) 0 0
\(790\) 12.9200 0.459674
\(791\) 54.2707 1.92964
\(792\) 0 0
\(793\) −0.436138 −0.0154877
\(794\) −0.450416 −0.0159847
\(795\) 0 0
\(796\) 22.3116 0.790815
\(797\) −13.4591 −0.476745 −0.238372 0.971174i \(-0.576614\pi\)
−0.238372 + 0.971174i \(0.576614\pi\)
\(798\) 0 0
\(799\) 5.53252 0.195726
\(800\) −14.9707 −0.529295
\(801\) 0 0
\(802\) −14.7263 −0.520002
\(803\) −14.1120 −0.498000
\(804\) 0 0
\(805\) 107.626 3.79333
\(806\) 0.557230 0.0196276
\(807\) 0 0
\(808\) −40.5489 −1.42651
\(809\) −29.8582 −1.04976 −0.524878 0.851177i \(-0.675889\pi\)
−0.524878 + 0.851177i \(0.675889\pi\)
\(810\) 0 0
\(811\) 23.1835 0.814083 0.407042 0.913410i \(-0.366560\pi\)
0.407042 + 0.913410i \(0.366560\pi\)
\(812\) 2.35156 0.0825236
\(813\) 0 0
\(814\) −1.74474 −0.0611531
\(815\) 37.0502 1.29781
\(816\) 0 0
\(817\) 2.60698 0.0912067
\(818\) 41.9688 1.46740
\(819\) 0 0
\(820\) 5.58749 0.195123
\(821\) −16.7615 −0.584980 −0.292490 0.956269i \(-0.594484\pi\)
−0.292490 + 0.956269i \(0.594484\pi\)
\(822\) 0 0
\(823\) −9.77979 −0.340902 −0.170451 0.985366i \(-0.554522\pi\)
−0.170451 + 0.985366i \(0.554522\pi\)
\(824\) −49.8445 −1.73641
\(825\) 0 0
\(826\) −3.48913 −0.121402
\(827\) 38.6934 1.34550 0.672751 0.739869i \(-0.265113\pi\)
0.672751 + 0.739869i \(0.265113\pi\)
\(828\) 0 0
\(829\) 7.90150 0.274430 0.137215 0.990541i \(-0.456185\pi\)
0.137215 + 0.990541i \(0.456185\pi\)
\(830\) −42.4644 −1.47396
\(831\) 0 0
\(832\) 2.65507 0.0920480
\(833\) 32.3784 1.12184
\(834\) 0 0
\(835\) −14.1167 −0.488529
\(836\) 1.39175 0.0481348
\(837\) 0 0
\(838\) 24.8391 0.858051
\(839\) 1.29253 0.0446230 0.0223115 0.999751i \(-0.492897\pi\)
0.0223115 + 0.999751i \(0.492897\pi\)
\(840\) 0 0
\(841\) −28.5913 −0.985906
\(842\) 17.2738 0.595294
\(843\) 0 0
\(844\) 3.45915 0.119069
\(845\) 37.0022 1.27291
\(846\) 0 0
\(847\) 39.5295 1.35825
\(848\) −0.0697021 −0.00239358
\(849\) 0 0
\(850\) −11.1624 −0.382866
\(851\) 12.8104 0.439134
\(852\) 0 0
\(853\) 47.2923 1.61926 0.809629 0.586942i \(-0.199668\pi\)
0.809629 + 0.586942i \(0.199668\pi\)
\(854\) −5.44621 −0.186366
\(855\) 0 0
\(856\) 32.9431 1.12597
\(857\) −41.5916 −1.42074 −0.710371 0.703827i \(-0.751473\pi\)
−0.710371 + 0.703827i \(0.751473\pi\)
\(858\) 0 0
\(859\) 39.2094 1.33781 0.668904 0.743349i \(-0.266764\pi\)
0.668904 + 0.743349i \(0.266764\pi\)
\(860\) −5.07410 −0.173025
\(861\) 0 0
\(862\) 25.4248 0.865972
\(863\) −37.3913 −1.27281 −0.636407 0.771354i \(-0.719580\pi\)
−0.636407 + 0.771354i \(0.719580\pi\)
\(864\) 0 0
\(865\) −44.6436 −1.51793
\(866\) 7.57864 0.257533
\(867\) 0 0
\(868\) −5.62596 −0.190957
\(869\) 5.04398 0.171105
\(870\) 0 0
\(871\) 1.38957 0.0470838
\(872\) 52.1751 1.76687
\(873\) 0 0
\(874\) 12.6387 0.427510
\(875\) −20.6438 −0.697889
\(876\) 0 0
\(877\) −42.5083 −1.43540 −0.717701 0.696351i \(-0.754806\pi\)
−0.717701 + 0.696351i \(0.754806\pi\)
\(878\) −22.2310 −0.750260
\(879\) 0 0
\(880\) 4.78535 0.161314
\(881\) −30.1193 −1.01475 −0.507373 0.861726i \(-0.669383\pi\)
−0.507373 + 0.861726i \(0.669383\pi\)
\(882\) 0 0
\(883\) −34.4505 −1.15935 −0.579676 0.814847i \(-0.696821\pi\)
−0.579676 + 0.814847i \(0.696821\pi\)
\(884\) −1.01068 −0.0339928
\(885\) 0 0
\(886\) −11.7264 −0.393956
\(887\) −47.0599 −1.58012 −0.790059 0.613031i \(-0.789950\pi\)
−0.790059 + 0.613031i \(0.789950\pi\)
\(888\) 0 0
\(889\) −53.6466 −1.79925
\(890\) 32.1899 1.07901
\(891\) 0 0
\(892\) −8.49748 −0.284517
\(893\) −2.23782 −0.0748859
\(894\) 0 0
\(895\) −48.7353 −1.62904
\(896\) −4.70705 −0.157252
\(897\) 0 0
\(898\) −2.23462 −0.0745701
\(899\) −0.977833 −0.0326125
\(900\) 0 0
\(901\) 0.161030 0.00536468
\(902\) −2.69796 −0.0898321
\(903\) 0 0
\(904\) −40.1503 −1.33538
\(905\) −9.48678 −0.315351
\(906\) 0 0
\(907\) 53.8831 1.78916 0.894579 0.446910i \(-0.147476\pi\)
0.894579 + 0.446910i \(0.147476\pi\)
\(908\) 7.10960 0.235940
\(909\) 0 0
\(910\) −4.30567 −0.142732
\(911\) −2.77566 −0.0919616 −0.0459808 0.998942i \(-0.514641\pi\)
−0.0459808 + 0.998942i \(0.514641\pi\)
\(912\) 0 0
\(913\) −16.5781 −0.548654
\(914\) −39.0484 −1.29161
\(915\) 0 0
\(916\) 10.0136 0.330857
\(917\) 84.5776 2.79300
\(918\) 0 0
\(919\) 19.6148 0.647031 0.323516 0.946223i \(-0.395135\pi\)
0.323516 + 0.946223i \(0.395135\pi\)
\(920\) −79.6237 −2.62512
\(921\) 0 0
\(922\) 31.4256 1.03495
\(923\) −0.125389 −0.00412724
\(924\) 0 0
\(925\) 4.57635 0.150469
\(926\) −21.4583 −0.705163
\(927\) 0 0
\(928\) −2.94197 −0.0965749
\(929\) −53.7834 −1.76458 −0.882289 0.470709i \(-0.843998\pi\)
−0.882289 + 0.470709i \(0.843998\pi\)
\(930\) 0 0
\(931\) −13.0966 −0.429224
\(932\) −14.0280 −0.459503
\(933\) 0 0
\(934\) 4.96914 0.162595
\(935\) −11.0554 −0.361550
\(936\) 0 0
\(937\) 31.1352 1.01714 0.508572 0.861020i \(-0.330174\pi\)
0.508572 + 0.861020i \(0.330174\pi\)
\(938\) 17.3521 0.566565
\(939\) 0 0
\(940\) 4.35559 0.142064
\(941\) −29.4881 −0.961286 −0.480643 0.876916i \(-0.659597\pi\)
−0.480643 + 0.876916i \(0.659597\pi\)
\(942\) 0 0
\(943\) 19.8092 0.645075
\(944\) 1.13905 0.0370730
\(945\) 0 0
\(946\) 2.45006 0.0796585
\(947\) 43.4174 1.41087 0.705437 0.708772i \(-0.250751\pi\)
0.705437 + 0.708772i \(0.250751\pi\)
\(948\) 0 0
\(949\) −4.14516 −0.134558
\(950\) 4.51502 0.146487
\(951\) 0 0
\(952\) −40.8510 −1.32399
\(953\) −14.4497 −0.468071 −0.234035 0.972228i \(-0.575193\pi\)
−0.234035 + 0.972228i \(0.575193\pi\)
\(954\) 0 0
\(955\) −35.1656 −1.13793
\(956\) −0.894124 −0.0289180
\(957\) 0 0
\(958\) 41.5308 1.34180
\(959\) 83.1247 2.68424
\(960\) 0 0
\(961\) −28.6606 −0.924535
\(962\) −0.512488 −0.0165233
\(963\) 0 0
\(964\) 22.8577 0.736196
\(965\) 44.0987 1.41959
\(966\) 0 0
\(967\) 6.88995 0.221566 0.110783 0.993845i \(-0.464664\pi\)
0.110783 + 0.993845i \(0.464664\pi\)
\(968\) −29.2446 −0.939956
\(969\) 0 0
\(970\) −30.8556 −0.990715
\(971\) 1.06821 0.0342806 0.0171403 0.999853i \(-0.494544\pi\)
0.0171403 + 0.999853i \(0.494544\pi\)
\(972\) 0 0
\(973\) −61.6640 −1.97686
\(974\) 27.4651 0.880037
\(975\) 0 0
\(976\) 1.77795 0.0569109
\(977\) 36.3679 1.16351 0.581755 0.813364i \(-0.302366\pi\)
0.581755 + 0.813364i \(0.302366\pi\)
\(978\) 0 0
\(979\) 12.5669 0.401640
\(980\) 25.4906 0.814266
\(981\) 0 0
\(982\) −21.1921 −0.676268
\(983\) −5.70415 −0.181934 −0.0909671 0.995854i \(-0.528996\pi\)
−0.0909671 + 0.995854i \(0.528996\pi\)
\(984\) 0 0
\(985\) −59.3974 −1.89256
\(986\) −2.19357 −0.0698575
\(987\) 0 0
\(988\) 0.408805 0.0130058
\(989\) −17.9891 −0.572019
\(990\) 0 0
\(991\) −6.98693 −0.221947 −0.110974 0.993823i \(-0.535397\pi\)
−0.110974 + 0.993823i \(0.535397\pi\)
\(992\) 7.03847 0.223472
\(993\) 0 0
\(994\) −1.56578 −0.0496636
\(995\) 71.6880 2.27266
\(996\) 0 0
\(997\) −28.4466 −0.900914 −0.450457 0.892798i \(-0.648739\pi\)
−0.450457 + 0.892798i \(0.648739\pi\)
\(998\) −17.4880 −0.553573
\(999\) 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.e.1.2 6
3.2 odd 2 717.2.a.d.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
717.2.a.d.1.5 6 3.2 odd 2
2151.2.a.e.1.2 6 1.1 even 1 trivial