Properties

Label 2151.2.a.k.1.19
Level $2151$
Weight $2$
Character 2151.1
Self dual yes
Analytic conductor $17.176$
Analytic rank $0$
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: \(0\)
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 1.19
Root \(2.61567\) of defining polynomial
Character \(\chi\) \(=\) 2151.1

$q$-expansion

\(f(q)\) \(=\) \(q+2.61567 q^{2} +4.84173 q^{4} +2.80971 q^{5} +3.73688 q^{7} +7.43303 q^{8} +O(q^{10})\) \(q+2.61567 q^{2} +4.84173 q^{4} +2.80971 q^{5} +3.73688 q^{7} +7.43303 q^{8} +7.34928 q^{10} -6.02355 q^{11} -0.651730 q^{13} +9.77444 q^{14} +9.75890 q^{16} +1.63490 q^{17} -6.09296 q^{19} +13.6039 q^{20} -15.7556 q^{22} +1.60504 q^{23} +2.89448 q^{25} -1.70471 q^{26} +18.0930 q^{28} +2.35372 q^{29} -6.90597 q^{31} +10.6600 q^{32} +4.27637 q^{34} +10.4995 q^{35} -10.0889 q^{37} -15.9372 q^{38} +20.8847 q^{40} +8.60185 q^{41} +8.78231 q^{43} -29.1644 q^{44} +4.19825 q^{46} -6.67446 q^{47} +6.96425 q^{49} +7.57100 q^{50} -3.15550 q^{52} -9.57641 q^{53} -16.9244 q^{55} +27.7763 q^{56} +6.15656 q^{58} -9.10915 q^{59} -9.50698 q^{61} -18.0637 q^{62} +8.36524 q^{64} -1.83117 q^{65} -8.35366 q^{67} +7.91576 q^{68} +27.4634 q^{70} +9.36039 q^{71} +8.06381 q^{73} -26.3893 q^{74} -29.5005 q^{76} -22.5092 q^{77} -4.58730 q^{79} +27.4197 q^{80} +22.4996 q^{82} +9.85175 q^{83} +4.59361 q^{85} +22.9716 q^{86} -44.7732 q^{88} +18.5981 q^{89} -2.43544 q^{91} +7.77117 q^{92} -17.4582 q^{94} -17.1195 q^{95} +6.39154 q^{97} +18.2162 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + O(q^{10}) \) \( 20q + 4q^{2} + 18q^{4} + 16q^{5} - 4q^{7} + 12q^{8} + 4q^{10} + 12q^{11} - 4q^{13} + 20q^{14} + 22q^{16} + 24q^{17} - 4q^{19} + 40q^{20} - 6q^{22} + 12q^{23} + 22q^{25} + 30q^{26} - 12q^{28} + 24q^{29} - 4q^{31} + 28q^{32} + 8q^{34} + 20q^{35} - 10q^{37} + 26q^{38} + 6q^{40} + 66q^{41} + 8q^{43} + 36q^{44} - 12q^{46} + 28q^{47} + 18q^{49} + 28q^{50} - 18q^{52} + 28q^{53} - 4q^{55} + 60q^{56} + 54q^{59} - 4q^{61} + 20q^{62} + 22q^{64} + 42q^{65} + 12q^{67} + 12q^{68} + 20q^{70} + 36q^{71} + 14q^{73} - 50q^{76} + 8q^{77} - 12q^{79} + 88q^{80} - 8q^{82} + 20q^{83} + 4q^{85} + 18q^{86} - 10q^{88} + 130q^{89} - 6q^{91} - 46q^{92} - 26q^{94} - 2q^{97} + 12q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 2.61567 1.84956 0.924779 0.380504i \(-0.124250\pi\)
0.924779 + 0.380504i \(0.124250\pi\)
\(3\) 0 0
\(4\) 4.84173 2.42087
\(5\) 2.80971 1.25654 0.628271 0.777995i \(-0.283763\pi\)
0.628271 + 0.777995i \(0.283763\pi\)
\(6\) 0 0
\(7\) 3.73688 1.41241 0.706203 0.708009i \(-0.250406\pi\)
0.706203 + 0.708009i \(0.250406\pi\)
\(8\) 7.43303 2.62797
\(9\) 0 0
\(10\) 7.34928 2.32405
\(11\) −6.02355 −1.81617 −0.908084 0.418789i \(-0.862455\pi\)
−0.908084 + 0.418789i \(0.862455\pi\)
\(12\) 0 0
\(13\) −0.651730 −0.180757 −0.0903787 0.995907i \(-0.528808\pi\)
−0.0903787 + 0.995907i \(0.528808\pi\)
\(14\) 9.77444 2.61233
\(15\) 0 0
\(16\) 9.75890 2.43972
\(17\) 1.63490 0.396522 0.198261 0.980149i \(-0.436471\pi\)
0.198261 + 0.980149i \(0.436471\pi\)
\(18\) 0 0
\(19\) −6.09296 −1.39782 −0.698910 0.715209i \(-0.746331\pi\)
−0.698910 + 0.715209i \(0.746331\pi\)
\(20\) 13.6039 3.04192
\(21\) 0 0
\(22\) −15.7556 −3.35911
\(23\) 1.60504 0.334674 0.167337 0.985900i \(-0.446483\pi\)
0.167337 + 0.985900i \(0.446483\pi\)
\(24\) 0 0
\(25\) 2.89448 0.578896
\(26\) −1.70471 −0.334321
\(27\) 0 0
\(28\) 18.0930 3.41925
\(29\) 2.35372 0.437075 0.218537 0.975829i \(-0.429871\pi\)
0.218537 + 0.975829i \(0.429871\pi\)
\(30\) 0 0
\(31\) −6.90597 −1.24035 −0.620174 0.784464i \(-0.712938\pi\)
−0.620174 + 0.784464i \(0.712938\pi\)
\(32\) 10.6600 1.88444
\(33\) 0 0
\(34\) 4.27637 0.733391
\(35\) 10.4995 1.77475
\(36\) 0 0
\(37\) −10.0889 −1.65861 −0.829305 0.558796i \(-0.811264\pi\)
−0.829305 + 0.558796i \(0.811264\pi\)
\(38\) −15.9372 −2.58535
\(39\) 0 0
\(40\) 20.8847 3.30216
\(41\) 8.60185 1.34338 0.671691 0.740831i \(-0.265568\pi\)
0.671691 + 0.740831i \(0.265568\pi\)
\(42\) 0 0
\(43\) 8.78231 1.33929 0.669645 0.742681i \(-0.266446\pi\)
0.669645 + 0.742681i \(0.266446\pi\)
\(44\) −29.1644 −4.39670
\(45\) 0 0
\(46\) 4.19825 0.618999
\(47\) −6.67446 −0.973570 −0.486785 0.873522i \(-0.661830\pi\)
−0.486785 + 0.873522i \(0.661830\pi\)
\(48\) 0 0
\(49\) 6.96425 0.994893
\(50\) 7.57100 1.07070
\(51\) 0 0
\(52\) −3.15550 −0.437589
\(53\) −9.57641 −1.31542 −0.657711 0.753271i \(-0.728475\pi\)
−0.657711 + 0.753271i \(0.728475\pi\)
\(54\) 0 0
\(55\) −16.9244 −2.28209
\(56\) 27.7763 3.71177
\(57\) 0 0
\(58\) 6.15656 0.808396
\(59\) −9.10915 −1.18591 −0.592955 0.805235i \(-0.702039\pi\)
−0.592955 + 0.805235i \(0.702039\pi\)
\(60\) 0 0
\(61\) −9.50698 −1.21724 −0.608622 0.793460i \(-0.708277\pi\)
−0.608622 + 0.793460i \(0.708277\pi\)
\(62\) −18.0637 −2.29410
\(63\) 0 0
\(64\) 8.36524 1.04565
\(65\) −1.83117 −0.227129
\(66\) 0 0
\(67\) −8.35366 −1.02056 −0.510281 0.860008i \(-0.670459\pi\)
−0.510281 + 0.860008i \(0.670459\pi\)
\(68\) 7.91576 0.959927
\(69\) 0 0
\(70\) 27.4634 3.28250
\(71\) 9.36039 1.11087 0.555437 0.831558i \(-0.312551\pi\)
0.555437 + 0.831558i \(0.312551\pi\)
\(72\) 0 0
\(73\) 8.06381 0.943798 0.471899 0.881653i \(-0.343569\pi\)
0.471899 + 0.881653i \(0.343569\pi\)
\(74\) −26.3893 −3.06770
\(75\) 0 0
\(76\) −29.5005 −3.38394
\(77\) −22.5092 −2.56517
\(78\) 0 0
\(79\) −4.58730 −0.516111 −0.258056 0.966130i \(-0.583082\pi\)
−0.258056 + 0.966130i \(0.583082\pi\)
\(80\) 27.4197 3.06561
\(81\) 0 0
\(82\) 22.4996 2.48466
\(83\) 9.85175 1.08137 0.540685 0.841225i \(-0.318165\pi\)
0.540685 + 0.841225i \(0.318165\pi\)
\(84\) 0 0
\(85\) 4.59361 0.498247
\(86\) 22.9716 2.47709
\(87\) 0 0
\(88\) −44.7732 −4.77284
\(89\) 18.5981 1.97139 0.985697 0.168526i \(-0.0539009\pi\)
0.985697 + 0.168526i \(0.0539009\pi\)
\(90\) 0 0
\(91\) −2.43544 −0.255303
\(92\) 7.77117 0.810200
\(93\) 0 0
\(94\) −17.4582 −1.80067
\(95\) −17.1195 −1.75642
\(96\) 0 0
\(97\) 6.39154 0.648962 0.324481 0.945892i \(-0.394810\pi\)
0.324481 + 0.945892i \(0.394810\pi\)
\(98\) 18.2162 1.84011
\(99\) 0 0
\(100\) 14.0143 1.40143
\(101\) −1.17857 −0.117272 −0.0586359 0.998279i \(-0.518675\pi\)
−0.0586359 + 0.998279i \(0.518675\pi\)
\(102\) 0 0
\(103\) 2.70231 0.266267 0.133133 0.991098i \(-0.457496\pi\)
0.133133 + 0.991098i \(0.457496\pi\)
\(104\) −4.84433 −0.475026
\(105\) 0 0
\(106\) −25.0487 −2.43295
\(107\) 10.4572 1.01093 0.505466 0.862847i \(-0.331321\pi\)
0.505466 + 0.862847i \(0.331321\pi\)
\(108\) 0 0
\(109\) 20.7374 1.98628 0.993141 0.116920i \(-0.0373021\pi\)
0.993141 + 0.116920i \(0.0373021\pi\)
\(110\) −44.2687 −4.22086
\(111\) 0 0
\(112\) 36.4678 3.44588
\(113\) 5.76532 0.542356 0.271178 0.962529i \(-0.412587\pi\)
0.271178 + 0.962529i \(0.412587\pi\)
\(114\) 0 0
\(115\) 4.50970 0.420531
\(116\) 11.3961 1.05810
\(117\) 0 0
\(118\) −23.8265 −2.19341
\(119\) 6.10943 0.560051
\(120\) 0 0
\(121\) 25.2831 2.29846
\(122\) −24.8671 −2.25136
\(123\) 0 0
\(124\) −33.4368 −3.00272
\(125\) −5.91590 −0.529135
\(126\) 0 0
\(127\) 6.92357 0.614368 0.307184 0.951650i \(-0.400613\pi\)
0.307184 + 0.951650i \(0.400613\pi\)
\(128\) 0.560710 0.0495603
\(129\) 0 0
\(130\) −4.78975 −0.420089
\(131\) 8.85105 0.773319 0.386660 0.922222i \(-0.373629\pi\)
0.386660 + 0.922222i \(0.373629\pi\)
\(132\) 0 0
\(133\) −22.7686 −1.97429
\(134\) −21.8504 −1.88759
\(135\) 0 0
\(136\) 12.1523 1.04205
\(137\) 8.08553 0.690794 0.345397 0.938457i \(-0.387744\pi\)
0.345397 + 0.938457i \(0.387744\pi\)
\(138\) 0 0
\(139\) −6.34824 −0.538450 −0.269225 0.963077i \(-0.586768\pi\)
−0.269225 + 0.963077i \(0.586768\pi\)
\(140\) 50.8360 4.29642
\(141\) 0 0
\(142\) 24.4837 2.05463
\(143\) 3.92573 0.328286
\(144\) 0 0
\(145\) 6.61328 0.549203
\(146\) 21.0923 1.74561
\(147\) 0 0
\(148\) −48.8479 −4.01527
\(149\) −12.2210 −1.00118 −0.500592 0.865683i \(-0.666884\pi\)
−0.500592 + 0.865683i \(0.666884\pi\)
\(150\) 0 0
\(151\) −13.7944 −1.12257 −0.561286 0.827622i \(-0.689693\pi\)
−0.561286 + 0.827622i \(0.689693\pi\)
\(152\) −45.2892 −3.67344
\(153\) 0 0
\(154\) −58.8768 −4.74443
\(155\) −19.4038 −1.55855
\(156\) 0 0
\(157\) 18.1273 1.44671 0.723356 0.690475i \(-0.242598\pi\)
0.723356 + 0.690475i \(0.242598\pi\)
\(158\) −11.9989 −0.954578
\(159\) 0 0
\(160\) 29.9515 2.36787
\(161\) 5.99783 0.472696
\(162\) 0 0
\(163\) −4.78122 −0.374494 −0.187247 0.982313i \(-0.559956\pi\)
−0.187247 + 0.982313i \(0.559956\pi\)
\(164\) 41.6478 3.25215
\(165\) 0 0
\(166\) 25.7689 2.00006
\(167\) −5.39807 −0.417715 −0.208857 0.977946i \(-0.566974\pi\)
−0.208857 + 0.977946i \(0.566974\pi\)
\(168\) 0 0
\(169\) −12.5752 −0.967327
\(170\) 12.0154 0.921536
\(171\) 0 0
\(172\) 42.5216 3.24224
\(173\) 16.5400 1.25751 0.628757 0.777602i \(-0.283564\pi\)
0.628757 + 0.777602i \(0.283564\pi\)
\(174\) 0 0
\(175\) 10.8163 0.817637
\(176\) −58.7832 −4.43095
\(177\) 0 0
\(178\) 48.6465 3.64621
\(179\) 10.8315 0.809584 0.404792 0.914409i \(-0.367344\pi\)
0.404792 + 0.914409i \(0.367344\pi\)
\(180\) 0 0
\(181\) −4.87344 −0.362240 −0.181120 0.983461i \(-0.557972\pi\)
−0.181120 + 0.983461i \(0.557972\pi\)
\(182\) −6.37030 −0.472198
\(183\) 0 0
\(184\) 11.9303 0.879514
\(185\) −28.3470 −2.08411
\(186\) 0 0
\(187\) −9.84791 −0.720151
\(188\) −32.3159 −2.35688
\(189\) 0 0
\(190\) −44.7789 −3.24860
\(191\) −6.92857 −0.501333 −0.250667 0.968073i \(-0.580650\pi\)
−0.250667 + 0.968073i \(0.580650\pi\)
\(192\) 0 0
\(193\) 18.4673 1.32931 0.664654 0.747151i \(-0.268579\pi\)
0.664654 + 0.747151i \(0.268579\pi\)
\(194\) 16.7182 1.20029
\(195\) 0 0
\(196\) 33.7190 2.40850
\(197\) 10.4671 0.745753 0.372877 0.927881i \(-0.378371\pi\)
0.372877 + 0.927881i \(0.378371\pi\)
\(198\) 0 0
\(199\) −1.98162 −0.140473 −0.0702367 0.997530i \(-0.522375\pi\)
−0.0702367 + 0.997530i \(0.522375\pi\)
\(200\) 21.5148 1.52132
\(201\) 0 0
\(202\) −3.08274 −0.216901
\(203\) 8.79557 0.617328
\(204\) 0 0
\(205\) 24.1687 1.68802
\(206\) 7.06835 0.492476
\(207\) 0 0
\(208\) −6.36017 −0.440998
\(209\) 36.7012 2.53868
\(210\) 0 0
\(211\) −12.8748 −0.886335 −0.443167 0.896439i \(-0.646145\pi\)
−0.443167 + 0.896439i \(0.646145\pi\)
\(212\) −46.3664 −3.18446
\(213\) 0 0
\(214\) 27.3525 1.86978
\(215\) 24.6758 1.68287
\(216\) 0 0
\(217\) −25.8068 −1.75188
\(218\) 54.2422 3.67375
\(219\) 0 0
\(220\) −81.9435 −5.52463
\(221\) −1.06552 −0.0716743
\(222\) 0 0
\(223\) −7.79435 −0.521949 −0.260974 0.965346i \(-0.584044\pi\)
−0.260974 + 0.965346i \(0.584044\pi\)
\(224\) 39.8351 2.66159
\(225\) 0 0
\(226\) 15.0802 1.00312
\(227\) 6.83248 0.453488 0.226744 0.973954i \(-0.427192\pi\)
0.226744 + 0.973954i \(0.427192\pi\)
\(228\) 0 0
\(229\) −16.7361 −1.10595 −0.552976 0.833197i \(-0.686508\pi\)
−0.552976 + 0.833197i \(0.686508\pi\)
\(230\) 11.7959 0.777797
\(231\) 0 0
\(232\) 17.4953 1.14862
\(233\) −0.0111558 −0.000730842 0 −0.000365421 1.00000i \(-0.500116\pi\)
−0.000365421 1.00000i \(0.500116\pi\)
\(234\) 0 0
\(235\) −18.7533 −1.22333
\(236\) −44.1041 −2.87093
\(237\) 0 0
\(238\) 15.9803 1.03585
\(239\) 1.00000 0.0646846
\(240\) 0 0
\(241\) −30.3328 −1.95391 −0.976955 0.213447i \(-0.931531\pi\)
−0.976955 + 0.213447i \(0.931531\pi\)
\(242\) 66.1322 4.25114
\(243\) 0 0
\(244\) −46.0303 −2.94678
\(245\) 19.5675 1.25012
\(246\) 0 0
\(247\) 3.97097 0.252666
\(248\) −51.3323 −3.25960
\(249\) 0 0
\(250\) −15.4741 −0.978665
\(251\) 12.7475 0.804616 0.402308 0.915504i \(-0.368208\pi\)
0.402308 + 0.915504i \(0.368208\pi\)
\(252\) 0 0
\(253\) −9.66803 −0.607824
\(254\) 18.1098 1.13631
\(255\) 0 0
\(256\) −15.2638 −0.953990
\(257\) 13.4247 0.837407 0.418704 0.908123i \(-0.362485\pi\)
0.418704 + 0.908123i \(0.362485\pi\)
\(258\) 0 0
\(259\) −37.7011 −2.34263
\(260\) −8.86605 −0.549849
\(261\) 0 0
\(262\) 23.1514 1.43030
\(263\) 6.07882 0.374836 0.187418 0.982280i \(-0.439988\pi\)
0.187418 + 0.982280i \(0.439988\pi\)
\(264\) 0 0
\(265\) −26.9070 −1.65288
\(266\) −59.5553 −3.65157
\(267\) 0 0
\(268\) −40.4462 −2.47064
\(269\) −28.1289 −1.71505 −0.857526 0.514441i \(-0.828000\pi\)
−0.857526 + 0.514441i \(0.828000\pi\)
\(270\) 0 0
\(271\) −8.81586 −0.535526 −0.267763 0.963485i \(-0.586284\pi\)
−0.267763 + 0.963485i \(0.586284\pi\)
\(272\) 15.9549 0.967405
\(273\) 0 0
\(274\) 21.1491 1.27766
\(275\) −17.4350 −1.05137
\(276\) 0 0
\(277\) −4.90099 −0.294472 −0.147236 0.989101i \(-0.547038\pi\)
−0.147236 + 0.989101i \(0.547038\pi\)
\(278\) −16.6049 −0.995895
\(279\) 0 0
\(280\) 78.0435 4.66399
\(281\) 13.1553 0.784780 0.392390 0.919799i \(-0.371648\pi\)
0.392390 + 0.919799i \(0.371648\pi\)
\(282\) 0 0
\(283\) −25.9060 −1.53995 −0.769976 0.638073i \(-0.779732\pi\)
−0.769976 + 0.638073i \(0.779732\pi\)
\(284\) 45.3205 2.68928
\(285\) 0 0
\(286\) 10.2684 0.607183
\(287\) 32.1440 1.89740
\(288\) 0 0
\(289\) −14.3271 −0.842770
\(290\) 17.2982 1.01578
\(291\) 0 0
\(292\) 39.0428 2.28481
\(293\) −6.11106 −0.357012 −0.178506 0.983939i \(-0.557126\pi\)
−0.178506 + 0.983939i \(0.557126\pi\)
\(294\) 0 0
\(295\) −25.5941 −1.49015
\(296\) −74.9914 −4.35879
\(297\) 0 0
\(298\) −31.9661 −1.85175
\(299\) −1.04605 −0.0604948
\(300\) 0 0
\(301\) 32.8184 1.89162
\(302\) −36.0816 −2.07626
\(303\) 0 0
\(304\) −59.4606 −3.41030
\(305\) −26.7119 −1.52952
\(306\) 0 0
\(307\) 22.7608 1.29903 0.649514 0.760350i \(-0.274972\pi\)
0.649514 + 0.760350i \(0.274972\pi\)
\(308\) −108.984 −6.20992
\(309\) 0 0
\(310\) −50.7539 −2.88263
\(311\) −0.0355362 −0.00201507 −0.00100754 0.999999i \(-0.500321\pi\)
−0.00100754 + 0.999999i \(0.500321\pi\)
\(312\) 0 0
\(313\) 19.1138 1.08038 0.540188 0.841544i \(-0.318353\pi\)
0.540188 + 0.841544i \(0.318353\pi\)
\(314\) 47.4149 2.67578
\(315\) 0 0
\(316\) −22.2105 −1.24944
\(317\) −18.9213 −1.06273 −0.531364 0.847143i \(-0.678320\pi\)
−0.531364 + 0.847143i \(0.678320\pi\)
\(318\) 0 0
\(319\) −14.1777 −0.793801
\(320\) 23.5039 1.31391
\(321\) 0 0
\(322\) 15.6884 0.874278
\(323\) −9.96140 −0.554267
\(324\) 0 0
\(325\) −1.88642 −0.104640
\(326\) −12.5061 −0.692649
\(327\) 0 0
\(328\) 63.9378 3.53037
\(329\) −24.9416 −1.37508
\(330\) 0 0
\(331\) −8.15827 −0.448419 −0.224209 0.974541i \(-0.571980\pi\)
−0.224209 + 0.974541i \(0.571980\pi\)
\(332\) 47.6995 2.61785
\(333\) 0 0
\(334\) −14.1196 −0.772588
\(335\) −23.4714 −1.28238
\(336\) 0 0
\(337\) 12.7368 0.693816 0.346908 0.937899i \(-0.387232\pi\)
0.346908 + 0.937899i \(0.387232\pi\)
\(338\) −32.8927 −1.78913
\(339\) 0 0
\(340\) 22.2410 1.20619
\(341\) 41.5984 2.25268
\(342\) 0 0
\(343\) −0.133596 −0.00721352
\(344\) 65.2792 3.51962
\(345\) 0 0
\(346\) 43.2632 2.32584
\(347\) 17.4737 0.938037 0.469019 0.883188i \(-0.344608\pi\)
0.469019 + 0.883188i \(0.344608\pi\)
\(348\) 0 0
\(349\) 17.3324 0.927781 0.463890 0.885893i \(-0.346453\pi\)
0.463890 + 0.885893i \(0.346453\pi\)
\(350\) 28.2919 1.51227
\(351\) 0 0
\(352\) −64.2110 −3.42246
\(353\) 9.60581 0.511266 0.255633 0.966774i \(-0.417716\pi\)
0.255633 + 0.966774i \(0.417716\pi\)
\(354\) 0 0
\(355\) 26.3000 1.39586
\(356\) 90.0470 4.77248
\(357\) 0 0
\(358\) 28.3316 1.49737
\(359\) −16.4121 −0.866199 −0.433099 0.901346i \(-0.642580\pi\)
−0.433099 + 0.901346i \(0.642580\pi\)
\(360\) 0 0
\(361\) 18.1242 0.953903
\(362\) −12.7473 −0.669984
\(363\) 0 0
\(364\) −11.7917 −0.618054
\(365\) 22.6570 1.18592
\(366\) 0 0
\(367\) 18.9634 0.989883 0.494942 0.868926i \(-0.335189\pi\)
0.494942 + 0.868926i \(0.335189\pi\)
\(368\) 15.6634 0.816512
\(369\) 0 0
\(370\) −74.1464 −3.85469
\(371\) −35.7859 −1.85791
\(372\) 0 0
\(373\) −16.6031 −0.859677 −0.429838 0.902906i \(-0.641429\pi\)
−0.429838 + 0.902906i \(0.641429\pi\)
\(374\) −25.7589 −1.33196
\(375\) 0 0
\(376\) −49.6115 −2.55852
\(377\) −1.53399 −0.0790045
\(378\) 0 0
\(379\) −1.05770 −0.0543306 −0.0271653 0.999631i \(-0.508648\pi\)
−0.0271653 + 0.999631i \(0.508648\pi\)
\(380\) −82.8878 −4.25206
\(381\) 0 0
\(382\) −18.1228 −0.927245
\(383\) 24.6868 1.26144 0.630719 0.776011i \(-0.282760\pi\)
0.630719 + 0.776011i \(0.282760\pi\)
\(384\) 0 0
\(385\) −63.2445 −3.22324
\(386\) 48.3045 2.45863
\(387\) 0 0
\(388\) 30.9461 1.57105
\(389\) −15.9721 −0.809818 −0.404909 0.914357i \(-0.632697\pi\)
−0.404909 + 0.914357i \(0.632697\pi\)
\(390\) 0 0
\(391\) 2.62408 0.132706
\(392\) 51.7655 2.61455
\(393\) 0 0
\(394\) 27.3786 1.37931
\(395\) −12.8890 −0.648515
\(396\) 0 0
\(397\) 11.2477 0.564505 0.282252 0.959340i \(-0.408918\pi\)
0.282252 + 0.959340i \(0.408918\pi\)
\(398\) −5.18327 −0.259814
\(399\) 0 0
\(400\) 28.2469 1.41235
\(401\) −15.0646 −0.752292 −0.376146 0.926560i \(-0.622751\pi\)
−0.376146 + 0.926560i \(0.622751\pi\)
\(402\) 0 0
\(403\) 4.50083 0.224202
\(404\) −5.70630 −0.283899
\(405\) 0 0
\(406\) 23.0063 1.14178
\(407\) 60.7712 3.01231
\(408\) 0 0
\(409\) −4.66906 −0.230870 −0.115435 0.993315i \(-0.536826\pi\)
−0.115435 + 0.993315i \(0.536826\pi\)
\(410\) 63.2174 3.12208
\(411\) 0 0
\(412\) 13.0839 0.644596
\(413\) −34.0398 −1.67499
\(414\) 0 0
\(415\) 27.6806 1.35879
\(416\) −6.94744 −0.340626
\(417\) 0 0
\(418\) 95.9983 4.69543
\(419\) −11.9766 −0.585095 −0.292547 0.956251i \(-0.594503\pi\)
−0.292547 + 0.956251i \(0.594503\pi\)
\(420\) 0 0
\(421\) 5.13637 0.250331 0.125166 0.992136i \(-0.460054\pi\)
0.125166 + 0.992136i \(0.460054\pi\)
\(422\) −33.6761 −1.63933
\(423\) 0 0
\(424\) −71.1818 −3.45689
\(425\) 4.73219 0.229545
\(426\) 0 0
\(427\) −35.5264 −1.71924
\(428\) 50.6307 2.44733
\(429\) 0 0
\(430\) 64.5437 3.11257
\(431\) 32.7602 1.57801 0.789003 0.614390i \(-0.210598\pi\)
0.789003 + 0.614390i \(0.210598\pi\)
\(432\) 0 0
\(433\) −32.9522 −1.58358 −0.791792 0.610791i \(-0.790852\pi\)
−0.791792 + 0.610791i \(0.790852\pi\)
\(434\) −67.5020 −3.24020
\(435\) 0 0
\(436\) 100.405 4.80852
\(437\) −9.77944 −0.467814
\(438\) 0 0
\(439\) 14.1383 0.674786 0.337393 0.941364i \(-0.390455\pi\)
0.337393 + 0.941364i \(0.390455\pi\)
\(440\) −125.800 −5.99727
\(441\) 0 0
\(442\) −2.78704 −0.132566
\(443\) −4.55179 −0.216262 −0.108131 0.994137i \(-0.534487\pi\)
−0.108131 + 0.994137i \(0.534487\pi\)
\(444\) 0 0
\(445\) 52.2553 2.47714
\(446\) −20.3875 −0.965374
\(447\) 0 0
\(448\) 31.2599 1.47689
\(449\) −19.4088 −0.915956 −0.457978 0.888963i \(-0.651426\pi\)
−0.457978 + 0.888963i \(0.651426\pi\)
\(450\) 0 0
\(451\) −51.8136 −2.43981
\(452\) 27.9141 1.31297
\(453\) 0 0
\(454\) 17.8715 0.838752
\(455\) −6.84287 −0.320799
\(456\) 0 0
\(457\) 23.2730 1.08866 0.544332 0.838870i \(-0.316783\pi\)
0.544332 + 0.838870i \(0.316783\pi\)
\(458\) −43.7761 −2.04552
\(459\) 0 0
\(460\) 21.8347 1.01805
\(461\) −20.7529 −0.966558 −0.483279 0.875466i \(-0.660554\pi\)
−0.483279 + 0.875466i \(0.660554\pi\)
\(462\) 0 0
\(463\) −21.0156 −0.976678 −0.488339 0.872654i \(-0.662397\pi\)
−0.488339 + 0.872654i \(0.662397\pi\)
\(464\) 22.9697 1.06634
\(465\) 0 0
\(466\) −0.0291799 −0.00135173
\(467\) −14.2561 −0.659693 −0.329846 0.944035i \(-0.606997\pi\)
−0.329846 + 0.944035i \(0.606997\pi\)
\(468\) 0 0
\(469\) −31.2166 −1.44145
\(470\) −49.0525 −2.26262
\(471\) 0 0
\(472\) −67.7086 −3.11654
\(473\) −52.9007 −2.43237
\(474\) 0 0
\(475\) −17.6359 −0.809193
\(476\) 29.5802 1.35581
\(477\) 0 0
\(478\) 2.61567 0.119638
\(479\) −0.790719 −0.0361289 −0.0180644 0.999837i \(-0.505750\pi\)
−0.0180644 + 0.999837i \(0.505750\pi\)
\(480\) 0 0
\(481\) 6.57526 0.299806
\(482\) −79.3407 −3.61387
\(483\) 0 0
\(484\) 122.414 5.56427
\(485\) 17.9584 0.815448
\(486\) 0 0
\(487\) −28.5642 −1.29437 −0.647184 0.762333i \(-0.724054\pi\)
−0.647184 + 0.762333i \(0.724054\pi\)
\(488\) −70.6657 −3.19889
\(489\) 0 0
\(490\) 51.1822 2.31218
\(491\) −41.2183 −1.86016 −0.930079 0.367361i \(-0.880261\pi\)
−0.930079 + 0.367361i \(0.880261\pi\)
\(492\) 0 0
\(493\) 3.84811 0.173310
\(494\) 10.3867 0.467321
\(495\) 0 0
\(496\) −67.3946 −3.02611
\(497\) 34.9786 1.56901
\(498\) 0 0
\(499\) −4.13535 −0.185124 −0.0925619 0.995707i \(-0.529506\pi\)
−0.0925619 + 0.995707i \(0.529506\pi\)
\(500\) −28.6432 −1.28096
\(501\) 0 0
\(502\) 33.3433 1.48818
\(503\) 26.8822 1.19862 0.599309 0.800518i \(-0.295442\pi\)
0.599309 + 0.800518i \(0.295442\pi\)
\(504\) 0 0
\(505\) −3.31143 −0.147357
\(506\) −25.2884 −1.12421
\(507\) 0 0
\(508\) 33.5221 1.48730
\(509\) 20.0950 0.890696 0.445348 0.895358i \(-0.353080\pi\)
0.445348 + 0.895358i \(0.353080\pi\)
\(510\) 0 0
\(511\) 30.1335 1.33303
\(512\) −41.0466 −1.81402
\(513\) 0 0
\(514\) 35.1145 1.54883
\(515\) 7.59271 0.334575
\(516\) 0 0
\(517\) 40.2039 1.76817
\(518\) −98.6137 −4.33284
\(519\) 0 0
\(520\) −13.6112 −0.596889
\(521\) 28.1803 1.23460 0.617302 0.786727i \(-0.288226\pi\)
0.617302 + 0.786727i \(0.288226\pi\)
\(522\) 0 0
\(523\) −4.66666 −0.204059 −0.102029 0.994781i \(-0.532534\pi\)
−0.102029 + 0.994781i \(0.532534\pi\)
\(524\) 42.8544 1.87210
\(525\) 0 0
\(526\) 15.9002 0.693281
\(527\) −11.2906 −0.491826
\(528\) 0 0
\(529\) −20.4238 −0.887993
\(530\) −70.3797 −3.05710
\(531\) 0 0
\(532\) −110.240 −4.77949
\(533\) −5.60608 −0.242826
\(534\) 0 0
\(535\) 29.3816 1.27028
\(536\) −62.0930 −2.68201
\(537\) 0 0
\(538\) −73.5760 −3.17209
\(539\) −41.9495 −1.80689
\(540\) 0 0
\(541\) −25.0475 −1.07687 −0.538437 0.842666i \(-0.680985\pi\)
−0.538437 + 0.842666i \(0.680985\pi\)
\(542\) −23.0594 −0.990486
\(543\) 0 0
\(544\) 17.4281 0.747222
\(545\) 58.2661 2.49585
\(546\) 0 0
\(547\) 41.9321 1.79289 0.896445 0.443156i \(-0.146141\pi\)
0.896445 + 0.443156i \(0.146141\pi\)
\(548\) 39.1480 1.67232
\(549\) 0 0
\(550\) −45.6043 −1.94457
\(551\) −14.3411 −0.610952
\(552\) 0 0
\(553\) −17.1422 −0.728959
\(554\) −12.8194 −0.544643
\(555\) 0 0
\(556\) −30.7364 −1.30352
\(557\) 1.75216 0.0742414 0.0371207 0.999311i \(-0.488181\pi\)
0.0371207 + 0.999311i \(0.488181\pi\)
\(558\) 0 0
\(559\) −5.72370 −0.242087
\(560\) 102.464 4.32989
\(561\) 0 0
\(562\) 34.4100 1.45150
\(563\) 42.8126 1.80434 0.902168 0.431385i \(-0.141975\pi\)
0.902168 + 0.431385i \(0.141975\pi\)
\(564\) 0 0
\(565\) 16.1989 0.681492
\(566\) −67.7615 −2.84823
\(567\) 0 0
\(568\) 69.5761 2.91935
\(569\) 4.33539 0.181749 0.0908745 0.995862i \(-0.471034\pi\)
0.0908745 + 0.995862i \(0.471034\pi\)
\(570\) 0 0
\(571\) 19.8015 0.828669 0.414334 0.910125i \(-0.364014\pi\)
0.414334 + 0.910125i \(0.364014\pi\)
\(572\) 19.0073 0.794735
\(573\) 0 0
\(574\) 84.0782 3.50936
\(575\) 4.64575 0.193741
\(576\) 0 0
\(577\) 6.90635 0.287515 0.143757 0.989613i \(-0.454081\pi\)
0.143757 + 0.989613i \(0.454081\pi\)
\(578\) −37.4749 −1.55875
\(579\) 0 0
\(580\) 32.0197 1.32955
\(581\) 36.8148 1.52733
\(582\) 0 0
\(583\) 57.6839 2.38902
\(584\) 59.9386 2.48028
\(585\) 0 0
\(586\) −15.9845 −0.660315
\(587\) 22.6076 0.933117 0.466558 0.884490i \(-0.345494\pi\)
0.466558 + 0.884490i \(0.345494\pi\)
\(588\) 0 0
\(589\) 42.0778 1.73378
\(590\) −66.9457 −2.75611
\(591\) 0 0
\(592\) −98.4569 −4.04655
\(593\) 32.5567 1.33694 0.668472 0.743737i \(-0.266948\pi\)
0.668472 + 0.743737i \(0.266948\pi\)
\(594\) 0 0
\(595\) 17.1657 0.703727
\(596\) −59.1709 −2.42373
\(597\) 0 0
\(598\) −2.73613 −0.111889
\(599\) −11.6681 −0.476746 −0.238373 0.971174i \(-0.576614\pi\)
−0.238373 + 0.971174i \(0.576614\pi\)
\(600\) 0 0
\(601\) −37.3270 −1.52260 −0.761301 0.648399i \(-0.775439\pi\)
−0.761301 + 0.648399i \(0.775439\pi\)
\(602\) 85.8422 3.49867
\(603\) 0 0
\(604\) −66.7888 −2.71760
\(605\) 71.0382 2.88811
\(606\) 0 0
\(607\) 5.15815 0.209363 0.104682 0.994506i \(-0.466618\pi\)
0.104682 + 0.994506i \(0.466618\pi\)
\(608\) −64.9509 −2.63411
\(609\) 0 0
\(610\) −69.8695 −2.82893
\(611\) 4.34995 0.175980
\(612\) 0 0
\(613\) −27.0102 −1.09093 −0.545466 0.838133i \(-0.683647\pi\)
−0.545466 + 0.838133i \(0.683647\pi\)
\(614\) 59.5348 2.40263
\(615\) 0 0
\(616\) −167.312 −6.74119
\(617\) 43.4026 1.74732 0.873662 0.486533i \(-0.161739\pi\)
0.873662 + 0.486533i \(0.161739\pi\)
\(618\) 0 0
\(619\) −10.2805 −0.413209 −0.206605 0.978425i \(-0.566241\pi\)
−0.206605 + 0.978425i \(0.566241\pi\)
\(620\) −93.9479 −3.77304
\(621\) 0 0
\(622\) −0.0929509 −0.00372699
\(623\) 69.4988 2.78441
\(624\) 0 0
\(625\) −31.0944 −1.24378
\(626\) 49.9954 1.99822
\(627\) 0 0
\(628\) 87.7673 3.50230
\(629\) −16.4944 −0.657676
\(630\) 0 0
\(631\) −28.2438 −1.12437 −0.562185 0.827012i \(-0.690039\pi\)
−0.562185 + 0.827012i \(0.690039\pi\)
\(632\) −34.0975 −1.35633
\(633\) 0 0
\(634\) −49.4920 −1.96558
\(635\) 19.4532 0.771978
\(636\) 0 0
\(637\) −4.53881 −0.179834
\(638\) −37.0843 −1.46818
\(639\) 0 0
\(640\) 1.57543 0.0622745
\(641\) −19.7227 −0.779000 −0.389500 0.921026i \(-0.627352\pi\)
−0.389500 + 0.921026i \(0.627352\pi\)
\(642\) 0 0
\(643\) 33.4104 1.31758 0.658788 0.752328i \(-0.271069\pi\)
0.658788 + 0.752328i \(0.271069\pi\)
\(644\) 29.0399 1.14433
\(645\) 0 0
\(646\) −26.0557 −1.02515
\(647\) −43.6081 −1.71441 −0.857205 0.514976i \(-0.827801\pi\)
−0.857205 + 0.514976i \(0.827801\pi\)
\(648\) 0 0
\(649\) 54.8694 2.15381
\(650\) −4.93425 −0.193537
\(651\) 0 0
\(652\) −23.1494 −0.906600
\(653\) −34.6094 −1.35437 −0.677186 0.735812i \(-0.736801\pi\)
−0.677186 + 0.735812i \(0.736801\pi\)
\(654\) 0 0
\(655\) 24.8689 0.971708
\(656\) 83.9445 3.27748
\(657\) 0 0
\(658\) −65.2391 −2.54328
\(659\) −40.3435 −1.57156 −0.785779 0.618507i \(-0.787738\pi\)
−0.785779 + 0.618507i \(0.787738\pi\)
\(660\) 0 0
\(661\) −5.19152 −0.201927 −0.100963 0.994890i \(-0.532192\pi\)
−0.100963 + 0.994890i \(0.532192\pi\)
\(662\) −21.3393 −0.829377
\(663\) 0 0
\(664\) 73.2284 2.84181
\(665\) −63.9733 −2.48078
\(666\) 0 0
\(667\) 3.77781 0.146278
\(668\) −26.1360 −1.01123
\(669\) 0 0
\(670\) −61.3934 −2.37183
\(671\) 57.2657 2.21072
\(672\) 0 0
\(673\) 50.4690 1.94544 0.972719 0.231987i \(-0.0745226\pi\)
0.972719 + 0.231987i \(0.0745226\pi\)
\(674\) 33.3152 1.28325
\(675\) 0 0
\(676\) −60.8860 −2.34177
\(677\) −20.6464 −0.793506 −0.396753 0.917925i \(-0.629863\pi\)
−0.396753 + 0.917925i \(0.629863\pi\)
\(678\) 0 0
\(679\) 23.8844 0.916599
\(680\) 34.1444 1.30938
\(681\) 0 0
\(682\) 108.808 4.16646
\(683\) −40.6049 −1.55370 −0.776852 0.629684i \(-0.783184\pi\)
−0.776852 + 0.629684i \(0.783184\pi\)
\(684\) 0 0
\(685\) 22.7180 0.868011
\(686\) −0.349444 −0.0133418
\(687\) 0 0
\(688\) 85.7057 3.26750
\(689\) 6.24124 0.237772
\(690\) 0 0
\(691\) −16.2838 −0.619463 −0.309732 0.950824i \(-0.600239\pi\)
−0.309732 + 0.950824i \(0.600239\pi\)
\(692\) 80.0823 3.04427
\(693\) 0 0
\(694\) 45.7054 1.73495
\(695\) −17.8367 −0.676585
\(696\) 0 0
\(697\) 14.0632 0.532681
\(698\) 45.3358 1.71598
\(699\) 0 0
\(700\) 52.3697 1.97939
\(701\) 46.6948 1.76364 0.881819 0.471589i \(-0.156319\pi\)
0.881819 + 0.471589i \(0.156319\pi\)
\(702\) 0 0
\(703\) 61.4715 2.31844
\(704\) −50.3884 −1.89908
\(705\) 0 0
\(706\) 25.1256 0.945616
\(707\) −4.40416 −0.165635
\(708\) 0 0
\(709\) −34.6969 −1.30307 −0.651535 0.758619i \(-0.725875\pi\)
−0.651535 + 0.758619i \(0.725875\pi\)
\(710\) 68.7921 2.58172
\(711\) 0 0
\(712\) 138.240 5.18077
\(713\) −11.0843 −0.415112
\(714\) 0 0
\(715\) 11.0302 0.412505
\(716\) 52.4432 1.95989
\(717\) 0 0
\(718\) −42.9287 −1.60208
\(719\) −25.9025 −0.966002 −0.483001 0.875620i \(-0.660453\pi\)
−0.483001 + 0.875620i \(0.660453\pi\)
\(720\) 0 0
\(721\) 10.0982 0.376077
\(722\) 47.4068 1.76430
\(723\) 0 0
\(724\) −23.5959 −0.876934
\(725\) 6.81280 0.253021
\(726\) 0 0
\(727\) −3.79662 −0.140809 −0.0704044 0.997519i \(-0.522429\pi\)
−0.0704044 + 0.997519i \(0.522429\pi\)
\(728\) −18.1027 −0.670929
\(729\) 0 0
\(730\) 59.2632 2.19343
\(731\) 14.3582 0.531058
\(732\) 0 0
\(733\) 43.6187 1.61109 0.805546 0.592533i \(-0.201872\pi\)
0.805546 + 0.592533i \(0.201872\pi\)
\(734\) 49.6021 1.83085
\(735\) 0 0
\(736\) 17.1097 0.630672
\(737\) 50.3187 1.85351
\(738\) 0 0
\(739\) 16.0631 0.590889 0.295445 0.955360i \(-0.404532\pi\)
0.295445 + 0.955360i \(0.404532\pi\)
\(740\) −137.249 −5.04536
\(741\) 0 0
\(742\) −93.6040 −3.43631
\(743\) −3.03523 −0.111352 −0.0556759 0.998449i \(-0.517731\pi\)
−0.0556759 + 0.998449i \(0.517731\pi\)
\(744\) 0 0
\(745\) −34.3375 −1.25803
\(746\) −43.4283 −1.59002
\(747\) 0 0
\(748\) −47.6809 −1.74339
\(749\) 39.0771 1.42785
\(750\) 0 0
\(751\) 36.3757 1.32737 0.663684 0.748013i \(-0.268992\pi\)
0.663684 + 0.748013i \(0.268992\pi\)
\(752\) −65.1353 −2.37524
\(753\) 0 0
\(754\) −4.01241 −0.146124
\(755\) −38.7583 −1.41056
\(756\) 0 0
\(757\) 5.22971 0.190077 0.0950384 0.995474i \(-0.469703\pi\)
0.0950384 + 0.995474i \(0.469703\pi\)
\(758\) −2.76660 −0.100488
\(759\) 0 0
\(760\) −127.249 −4.61582
\(761\) 48.1497 1.74543 0.872713 0.488234i \(-0.162359\pi\)
0.872713 + 0.488234i \(0.162359\pi\)
\(762\) 0 0
\(763\) 77.4931 2.80544
\(764\) −33.5463 −1.21366
\(765\) 0 0
\(766\) 64.5726 2.33310
\(767\) 5.93671 0.214362
\(768\) 0 0
\(769\) 19.4642 0.701895 0.350948 0.936395i \(-0.385859\pi\)
0.350948 + 0.936395i \(0.385859\pi\)
\(770\) −165.427 −5.96157
\(771\) 0 0
\(772\) 89.4139 3.21808
\(773\) 45.5889 1.63972 0.819859 0.572566i \(-0.194052\pi\)
0.819859 + 0.572566i \(0.194052\pi\)
\(774\) 0 0
\(775\) −19.9892 −0.718033
\(776\) 47.5085 1.70546
\(777\) 0 0
\(778\) −41.7778 −1.49781
\(779\) −52.4107 −1.87781
\(780\) 0 0
\(781\) −56.3828 −2.01753
\(782\) 6.86374 0.245447
\(783\) 0 0
\(784\) 67.9634 2.42726
\(785\) 50.9324 1.81785
\(786\) 0 0
\(787\) 20.4385 0.728553 0.364277 0.931291i \(-0.381316\pi\)
0.364277 + 0.931291i \(0.381316\pi\)
\(788\) 50.6791 1.80537
\(789\) 0 0
\(790\) −33.7133 −1.19947
\(791\) 21.5443 0.766027
\(792\) 0 0
\(793\) 6.19599 0.220026
\(794\) 29.4202 1.04408
\(795\) 0 0
\(796\) −9.59447 −0.340067
\(797\) 2.96900 0.105167 0.0525837 0.998617i \(-0.483254\pi\)
0.0525837 + 0.998617i \(0.483254\pi\)
\(798\) 0 0
\(799\) −10.9121 −0.386042
\(800\) 30.8551 1.09089
\(801\) 0 0
\(802\) −39.4041 −1.39141
\(803\) −48.5728 −1.71410
\(804\) 0 0
\(805\) 16.8522 0.593961
\(806\) 11.7727 0.414675
\(807\) 0 0
\(808\) −8.76032 −0.308187
\(809\) −41.1587 −1.44706 −0.723531 0.690292i \(-0.757482\pi\)
−0.723531 + 0.690292i \(0.757482\pi\)
\(810\) 0 0
\(811\) 1.67476 0.0588087 0.0294044 0.999568i \(-0.490639\pi\)
0.0294044 + 0.999568i \(0.490639\pi\)
\(812\) 42.5858 1.49447
\(813\) 0 0
\(814\) 158.957 5.57145
\(815\) −13.4339 −0.470567
\(816\) 0 0
\(817\) −53.5103 −1.87209
\(818\) −12.2127 −0.427008
\(819\) 0 0
\(820\) 117.018 4.08646
\(821\) 3.51494 0.122672 0.0613362 0.998117i \(-0.480464\pi\)
0.0613362 + 0.998117i \(0.480464\pi\)
\(822\) 0 0
\(823\) −28.6281 −0.997914 −0.498957 0.866627i \(-0.666284\pi\)
−0.498957 + 0.866627i \(0.666284\pi\)
\(824\) 20.0864 0.699741
\(825\) 0 0
\(826\) −89.0369 −3.09799
\(827\) −29.3296 −1.01989 −0.509945 0.860207i \(-0.670334\pi\)
−0.509945 + 0.860207i \(0.670334\pi\)
\(828\) 0 0
\(829\) 18.4506 0.640817 0.320408 0.947280i \(-0.396180\pi\)
0.320408 + 0.947280i \(0.396180\pi\)
\(830\) 72.4033 2.51315
\(831\) 0 0
\(832\) −5.45188 −0.189010
\(833\) 11.3859 0.394497
\(834\) 0 0
\(835\) −15.1670 −0.524876
\(836\) 177.697 6.14579
\(837\) 0 0
\(838\) −31.3268 −1.08217
\(839\) 13.8785 0.479140 0.239570 0.970879i \(-0.422994\pi\)
0.239570 + 0.970879i \(0.422994\pi\)
\(840\) 0 0
\(841\) −23.4600 −0.808965
\(842\) 13.4351 0.463002
\(843\) 0 0
\(844\) −62.3361 −2.14570
\(845\) −35.3328 −1.21549
\(846\) 0 0
\(847\) 94.4798 3.24636
\(848\) −93.4552 −3.20927
\(849\) 0 0
\(850\) 12.3779 0.424557
\(851\) −16.1931 −0.555094
\(852\) 0 0
\(853\) −39.3799 −1.34834 −0.674171 0.738575i \(-0.735499\pi\)
−0.674171 + 0.738575i \(0.735499\pi\)
\(854\) −92.9254 −3.17984
\(855\) 0 0
\(856\) 77.7284 2.65670
\(857\) 10.7445 0.367026 0.183513 0.983017i \(-0.441253\pi\)
0.183513 + 0.983017i \(0.441253\pi\)
\(858\) 0 0
\(859\) −40.7933 −1.39185 −0.695925 0.718114i \(-0.745005\pi\)
−0.695925 + 0.718114i \(0.745005\pi\)
\(860\) 119.473 4.07401
\(861\) 0 0
\(862\) 85.6900 2.91861
\(863\) −14.6436 −0.498475 −0.249238 0.968442i \(-0.580180\pi\)
−0.249238 + 0.968442i \(0.580180\pi\)
\(864\) 0 0
\(865\) 46.4726 1.58012
\(866\) −86.1922 −2.92893
\(867\) 0 0
\(868\) −124.949 −4.24106
\(869\) 27.6318 0.937345
\(870\) 0 0
\(871\) 5.44433 0.184474
\(872\) 154.142 5.21990
\(873\) 0 0
\(874\) −25.5798 −0.865249
\(875\) −22.1070 −0.747353
\(876\) 0 0
\(877\) 2.46280 0.0831628 0.0415814 0.999135i \(-0.486760\pi\)
0.0415814 + 0.999135i \(0.486760\pi\)
\(878\) 36.9812 1.24806
\(879\) 0 0
\(880\) −165.164 −5.56767
\(881\) −18.9071 −0.636996 −0.318498 0.947924i \(-0.603178\pi\)
−0.318498 + 0.947924i \(0.603178\pi\)
\(882\) 0 0
\(883\) 25.7316 0.865938 0.432969 0.901409i \(-0.357466\pi\)
0.432969 + 0.901409i \(0.357466\pi\)
\(884\) −5.15894 −0.173514
\(885\) 0 0
\(886\) −11.9060 −0.399990
\(887\) −9.26612 −0.311126 −0.155563 0.987826i \(-0.549719\pi\)
−0.155563 + 0.987826i \(0.549719\pi\)
\(888\) 0 0
\(889\) 25.8725 0.867737
\(890\) 136.683 4.58161
\(891\) 0 0
\(892\) −37.7382 −1.26357
\(893\) 40.6672 1.36088
\(894\) 0 0
\(895\) 30.4334 1.01728
\(896\) 2.09531 0.0699993
\(897\) 0 0
\(898\) −50.7669 −1.69411
\(899\) −16.2547 −0.542125
\(900\) 0 0
\(901\) −15.6565 −0.521594
\(902\) −135.527 −4.51257
\(903\) 0 0
\(904\) 42.8538 1.42530
\(905\) −13.6930 −0.455170
\(906\) 0 0
\(907\) −1.45986 −0.0484740 −0.0242370 0.999706i \(-0.507716\pi\)
−0.0242370 + 0.999706i \(0.507716\pi\)
\(908\) 33.0811 1.09783
\(909\) 0 0
\(910\) −17.8987 −0.593336
\(911\) −40.8089 −1.35206 −0.676030 0.736874i \(-0.736301\pi\)
−0.676030 + 0.736874i \(0.736301\pi\)
\(912\) 0 0
\(913\) −59.3425 −1.96395
\(914\) 60.8745 2.01355
\(915\) 0 0
\(916\) −81.0317 −2.67736
\(917\) 33.0753 1.09224
\(918\) 0 0
\(919\) −10.1810 −0.335839 −0.167919 0.985801i \(-0.553705\pi\)
−0.167919 + 0.985801i \(0.553705\pi\)
\(920\) 33.5207 1.10515
\(921\) 0 0
\(922\) −54.2827 −1.78770
\(923\) −6.10045 −0.200799
\(924\) 0 0
\(925\) −29.2022 −0.960163
\(926\) −54.9699 −1.80642
\(927\) 0 0
\(928\) 25.0907 0.823641
\(929\) 6.89974 0.226373 0.113187 0.993574i \(-0.463894\pi\)
0.113187 + 0.993574i \(0.463894\pi\)
\(930\) 0 0
\(931\) −42.4329 −1.39068
\(932\) −0.0540135 −0.00176927
\(933\) 0 0
\(934\) −37.2892 −1.22014
\(935\) −27.6698 −0.904899
\(936\) 0 0
\(937\) 36.1115 1.17971 0.589856 0.807509i \(-0.299185\pi\)
0.589856 + 0.807509i \(0.299185\pi\)
\(938\) −81.6524 −2.66604
\(939\) 0 0
\(940\) −90.7984 −2.96152
\(941\) 37.6486 1.22731 0.613655 0.789574i \(-0.289698\pi\)
0.613655 + 0.789574i \(0.289698\pi\)
\(942\) 0 0
\(943\) 13.8063 0.449595
\(944\) −88.8953 −2.89330
\(945\) 0 0
\(946\) −138.371 −4.49882
\(947\) −7.65789 −0.248848 −0.124424 0.992229i \(-0.539708\pi\)
−0.124424 + 0.992229i \(0.539708\pi\)
\(948\) 0 0
\(949\) −5.25543 −0.170599
\(950\) −46.1298 −1.49665
\(951\) 0 0
\(952\) 45.4116 1.47180
\(953\) −3.61700 −0.117166 −0.0585830 0.998283i \(-0.518658\pi\)
−0.0585830 + 0.998283i \(0.518658\pi\)
\(954\) 0 0
\(955\) −19.4673 −0.629946
\(956\) 4.84173 0.156593
\(957\) 0 0
\(958\) −2.06826 −0.0668225
\(959\) 30.2146 0.975682
\(960\) 0 0
\(961\) 16.6924 0.538464
\(962\) 17.1987 0.554509
\(963\) 0 0
\(964\) −146.863 −4.73015
\(965\) 51.8879 1.67033
\(966\) 0 0
\(967\) 40.7102 1.30915 0.654576 0.755996i \(-0.272847\pi\)
0.654576 + 0.755996i \(0.272847\pi\)
\(968\) 187.930 6.04030
\(969\) 0 0
\(970\) 46.9732 1.50822
\(971\) 26.5443 0.851848 0.425924 0.904759i \(-0.359949\pi\)
0.425924 + 0.904759i \(0.359949\pi\)
\(972\) 0 0
\(973\) −23.7226 −0.760511
\(974\) −74.7146 −2.39401
\(975\) 0 0
\(976\) −92.7777 −2.96974
\(977\) 17.4506 0.558295 0.279147 0.960248i \(-0.409948\pi\)
0.279147 + 0.960248i \(0.409948\pi\)
\(978\) 0 0
\(979\) −112.026 −3.58038
\(980\) 94.7407 3.02638
\(981\) 0 0
\(982\) −107.814 −3.44047
\(983\) 46.7209 1.49016 0.745082 0.666973i \(-0.232410\pi\)
0.745082 + 0.666973i \(0.232410\pi\)
\(984\) 0 0
\(985\) 29.4097 0.937070
\(986\) 10.0654 0.320547
\(987\) 0 0
\(988\) 19.2263 0.611672
\(989\) 14.0960 0.448225
\(990\) 0 0
\(991\) −55.4287 −1.76075 −0.880375 0.474279i \(-0.842709\pi\)
−0.880375 + 0.474279i \(0.842709\pi\)
\(992\) −73.6176 −2.33736
\(993\) 0 0
\(994\) 91.4926 2.90197
\(995\) −5.56778 −0.176511
\(996\) 0 0
\(997\) 37.5805 1.19019 0.595093 0.803657i \(-0.297115\pi\)
0.595093 + 0.803657i \(0.297115\pi\)
\(998\) −10.8167 −0.342397
\(999\) 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.k.1.19 yes 20
3.2 odd 2 2151.2.a.j.1.2 20
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
2151.2.a.j.1.2 20 3.2 odd 2
2151.2.a.k.1.19 yes 20 1.1 even 1 trivial