Properties

Label 1183.2.a.p.1.2
Level $1183$
Weight $2$
Character 1183.1
Self dual yes
Analytic conductor $9.446$
Analytic rank $0$
Dimension $6$
CM no
Inner twists $1$

Related objects

Downloads

Learn more

Newspace parameters

Level: \( N \) \(=\) \( 1183 = 7 \cdot 13^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1183.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(9.44630255912\)
Analytic rank: \(0\)
Dimension: \(6\)
Coefficient field: 6.6.7674048.1
Defining polynomial: \( x^{6} - 2x^{5} - 5x^{4} + 8x^{3} + 7x^{2} - 6x - 2 \) Copy content Toggle raw display
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 91)
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.2
Root \(1.82356\) of defining polynomial
Character \(\chi\) \(=\) 1183.1

$q$-expansion

\(f(q)\) \(=\) \(q-0.823556 q^{2} +2.66029 q^{3} -1.32176 q^{4} +3.16209 q^{5} -2.19090 q^{6} -1.00000 q^{7} +2.73565 q^{8} +4.07715 q^{9} +O(q^{10})\) \(q-0.823556 q^{2} +2.66029 q^{3} -1.32176 q^{4} +3.16209 q^{5} -2.19090 q^{6} -1.00000 q^{7} +2.73565 q^{8} +4.07715 q^{9} -2.60416 q^{10} +5.94270 q^{11} -3.51626 q^{12} +0.823556 q^{14} +8.41209 q^{15} +0.390549 q^{16} -2.69964 q^{17} -3.35776 q^{18} +1.95705 q^{19} -4.17951 q^{20} -2.66029 q^{21} -4.89414 q^{22} -2.72941 q^{23} +7.27763 q^{24} +4.99883 q^{25} +2.86554 q^{27} +1.32176 q^{28} -5.99845 q^{29} -6.92783 q^{30} +1.15155 q^{31} -5.79294 q^{32} +15.8093 q^{33} +2.22331 q^{34} -3.16209 q^{35} -5.38900 q^{36} -6.50454 q^{37} -1.61174 q^{38} +8.65038 q^{40} +3.73374 q^{41} +2.19090 q^{42} +6.99125 q^{43} -7.85479 q^{44} +12.8923 q^{45} +2.24783 q^{46} +0.456071 q^{47} +1.03897 q^{48} +1.00000 q^{49} -4.11682 q^{50} -7.18184 q^{51} +0.399286 q^{53} -2.35994 q^{54} +18.7914 q^{55} -2.73565 q^{56} +5.20632 q^{57} +4.94006 q^{58} -4.80586 q^{59} -11.1187 q^{60} -1.15703 q^{61} -0.948365 q^{62} -4.07715 q^{63} +3.98971 q^{64} -13.0199 q^{66} +6.27918 q^{67} +3.56827 q^{68} -7.26104 q^{69} +2.60416 q^{70} +4.50720 q^{71} +11.1537 q^{72} +8.30575 q^{73} +5.35685 q^{74} +13.2983 q^{75} -2.58674 q^{76} -5.94270 q^{77} -7.91410 q^{79} +1.23495 q^{80} -4.60828 q^{81} -3.07494 q^{82} -6.19795 q^{83} +3.51626 q^{84} -8.53652 q^{85} -5.75769 q^{86} -15.9576 q^{87} +16.2571 q^{88} +3.56136 q^{89} -10.6176 q^{90} +3.60762 q^{92} +3.06345 q^{93} -0.375600 q^{94} +6.18837 q^{95} -15.4109 q^{96} -3.42751 q^{97} -0.823556 q^{98} +24.2293 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9}+O(q^{10}) \) Copy content Toggle raw display \( 6 q + 4 q^{2} + 4 q^{4} + 6 q^{5} + 4 q^{6} - 6 q^{7} + 12 q^{8} + 4 q^{9} + 12 q^{10} + 4 q^{11} + 2 q^{12} - 4 q^{14} + 20 q^{15} + 8 q^{16} - 4 q^{17} - 16 q^{18} + 2 q^{19} + 26 q^{20} - 6 q^{22} - 12 q^{23} + 2 q^{24} + 10 q^{25} + 6 q^{27} - 4 q^{28} - 8 q^{29} + 8 q^{30} - 14 q^{31} + 8 q^{32} + 16 q^{33} - 2 q^{34} - 6 q^{35} - 10 q^{36} + 12 q^{37} - 2 q^{38} + 46 q^{40} + 28 q^{41} - 4 q^{42} + 2 q^{43} - 20 q^{44} + 16 q^{45} + 20 q^{46} + 14 q^{47} + 2 q^{48} + 6 q^{49} + 32 q^{50} - 26 q^{51} - 22 q^{53} + 14 q^{54} + 6 q^{55} - 12 q^{56} + 4 q^{58} - 2 q^{59} - 14 q^{61} - 4 q^{62} - 4 q^{63} + 26 q^{64} - 26 q^{66} + 24 q^{67} + 8 q^{68} + 4 q^{69} - 12 q^{70} + 4 q^{71} + 8 q^{72} + 36 q^{73} - 6 q^{74} + 46 q^{75} - 26 q^{76} - 4 q^{77} - 28 q^{79} + 36 q^{80} - 2 q^{81} + 14 q^{82} + 26 q^{83} - 2 q^{84} - 20 q^{85} - 24 q^{86} + 2 q^{87} - 14 q^{88} + 42 q^{89} - 12 q^{90} + 12 q^{92} - 4 q^{94} - 22 q^{95} - 42 q^{96} + 24 q^{97} + 4 q^{98} + 16 q^{99}+O(q^{100}) \) Copy content Toggle raw display

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −0.823556 −0.582342 −0.291171 0.956671i \(-0.594045\pi\)
−0.291171 + 0.956671i \(0.594045\pi\)
\(3\) 2.66029 1.53592 0.767960 0.640498i \(-0.221272\pi\)
0.767960 + 0.640498i \(0.221272\pi\)
\(4\) −1.32176 −0.660878
\(5\) 3.16209 1.41413 0.707065 0.707148i \(-0.250019\pi\)
0.707065 + 0.707148i \(0.250019\pi\)
\(6\) −2.19090 −0.894431
\(7\) −1.00000 −0.377964
\(8\) 2.73565 0.967199
\(9\) 4.07715 1.35905
\(10\) −2.60416 −0.823508
\(11\) 5.94270 1.79179 0.895895 0.444265i \(-0.146535\pi\)
0.895895 + 0.444265i \(0.146535\pi\)
\(12\) −3.51626 −1.01506
\(13\) 0 0
\(14\) 0.823556 0.220105
\(15\) 8.41209 2.17199
\(16\) 0.390549 0.0976372
\(17\) −2.69964 −0.654760 −0.327380 0.944893i \(-0.606166\pi\)
−0.327380 + 0.944893i \(0.606166\pi\)
\(18\) −3.35776 −0.791433
\(19\) 1.95705 0.448978 0.224489 0.974477i \(-0.427929\pi\)
0.224489 + 0.974477i \(0.427929\pi\)
\(20\) −4.17951 −0.934568
\(21\) −2.66029 −0.580523
\(22\) −4.89414 −1.04343
\(23\) −2.72941 −0.569122 −0.284561 0.958658i \(-0.591848\pi\)
−0.284561 + 0.958658i \(0.591848\pi\)
\(24\) 7.27763 1.48554
\(25\) 4.99883 0.999766
\(26\) 0 0
\(27\) 2.86554 0.551474
\(28\) 1.32176 0.249788
\(29\) −5.99845 −1.11388 −0.556942 0.830551i \(-0.688026\pi\)
−0.556942 + 0.830551i \(0.688026\pi\)
\(30\) −6.92783 −1.26484
\(31\) 1.15155 0.206824 0.103412 0.994639i \(-0.467024\pi\)
0.103412 + 0.994639i \(0.467024\pi\)
\(32\) −5.79294 −1.02406
\(33\) 15.8093 2.75205
\(34\) 2.22331 0.381294
\(35\) −3.16209 −0.534491
\(36\) −5.38900 −0.898167
\(37\) −6.50454 −1.06934 −0.534670 0.845061i \(-0.679564\pi\)
−0.534670 + 0.845061i \(0.679564\pi\)
\(38\) −1.61174 −0.261459
\(39\) 0 0
\(40\) 8.65038 1.36775
\(41\) 3.73374 0.583112 0.291556 0.956554i \(-0.405827\pi\)
0.291556 + 0.956554i \(0.405827\pi\)
\(42\) 2.19090 0.338063
\(43\) 6.99125 1.06616 0.533078 0.846066i \(-0.321035\pi\)
0.533078 + 0.846066i \(0.321035\pi\)
\(44\) −7.85479 −1.18415
\(45\) 12.8923 1.92188
\(46\) 2.24783 0.331424
\(47\) 0.456071 0.0665248 0.0332624 0.999447i \(-0.489410\pi\)
0.0332624 + 0.999447i \(0.489410\pi\)
\(48\) 1.03897 0.149963
\(49\) 1.00000 0.142857
\(50\) −4.11682 −0.582206
\(51\) −7.18184 −1.00566
\(52\) 0 0
\(53\) 0.399286 0.0548462 0.0274231 0.999624i \(-0.491270\pi\)
0.0274231 + 0.999624i \(0.491270\pi\)
\(54\) −2.35994 −0.321146
\(55\) 18.7914 2.53383
\(56\) −2.73565 −0.365567
\(57\) 5.20632 0.689594
\(58\) 4.94006 0.648662
\(59\) −4.80586 −0.625670 −0.312835 0.949807i \(-0.601279\pi\)
−0.312835 + 0.949807i \(0.601279\pi\)
\(60\) −11.1187 −1.43542
\(61\) −1.15703 −0.148142 −0.0740711 0.997253i \(-0.523599\pi\)
−0.0740711 + 0.997253i \(0.523599\pi\)
\(62\) −0.948365 −0.120442
\(63\) −4.07715 −0.513673
\(64\) 3.98971 0.498714
\(65\) 0 0
\(66\) −13.0199 −1.60263
\(67\) 6.27918 0.767124 0.383562 0.923515i \(-0.374697\pi\)
0.383562 + 0.923515i \(0.374697\pi\)
\(68\) 3.56827 0.432716
\(69\) −7.26104 −0.874127
\(70\) 2.60416 0.311257
\(71\) 4.50720 0.534906 0.267453 0.963571i \(-0.413818\pi\)
0.267453 + 0.963571i \(0.413818\pi\)
\(72\) 11.1537 1.31447
\(73\) 8.30575 0.972115 0.486057 0.873927i \(-0.338435\pi\)
0.486057 + 0.873927i \(0.338435\pi\)
\(74\) 5.35685 0.622721
\(75\) 13.2983 1.53556
\(76\) −2.58674 −0.296719
\(77\) −5.94270 −0.677233
\(78\) 0 0
\(79\) −7.91410 −0.890405 −0.445203 0.895430i \(-0.646868\pi\)
−0.445203 + 0.895430i \(0.646868\pi\)
\(80\) 1.23495 0.138072
\(81\) −4.60828 −0.512031
\(82\) −3.07494 −0.339571
\(83\) −6.19795 −0.680313 −0.340156 0.940369i \(-0.610480\pi\)
−0.340156 + 0.940369i \(0.610480\pi\)
\(84\) 3.51626 0.383655
\(85\) −8.53652 −0.925916
\(86\) −5.75769 −0.620867
\(87\) −15.9576 −1.71084
\(88\) 16.2571 1.73302
\(89\) 3.56136 0.377504 0.188752 0.982025i \(-0.439556\pi\)
0.188752 + 0.982025i \(0.439556\pi\)
\(90\) −10.6176 −1.11919
\(91\) 0 0
\(92\) 3.60762 0.376120
\(93\) 3.06345 0.317665
\(94\) −0.375600 −0.0387402
\(95\) 6.18837 0.634913
\(96\) −15.4109 −1.57287
\(97\) −3.42751 −0.348011 −0.174005 0.984745i \(-0.555671\pi\)
−0.174005 + 0.984745i \(0.555671\pi\)
\(98\) −0.823556 −0.0831917
\(99\) 24.2293 2.43513
\(100\) −6.60723 −0.660723
\(101\) −13.3295 −1.32633 −0.663167 0.748472i \(-0.730788\pi\)
−0.663167 + 0.748472i \(0.730788\pi\)
\(102\) 5.91465 0.585637
\(103\) 11.6450 1.14741 0.573706 0.819061i \(-0.305505\pi\)
0.573706 + 0.819061i \(0.305505\pi\)
\(104\) 0 0
\(105\) −8.41209 −0.820936
\(106\) −0.328834 −0.0319392
\(107\) 3.92966 0.379894 0.189947 0.981794i \(-0.439168\pi\)
0.189947 + 0.981794i \(0.439168\pi\)
\(108\) −3.78755 −0.364457
\(109\) −11.2533 −1.07787 −0.538936 0.842346i \(-0.681174\pi\)
−0.538936 + 0.842346i \(0.681174\pi\)
\(110\) −15.4757 −1.47555
\(111\) −17.3040 −1.64242
\(112\) −0.390549 −0.0369034
\(113\) −5.77418 −0.543189 −0.271595 0.962412i \(-0.587551\pi\)
−0.271595 + 0.962412i \(0.587551\pi\)
\(114\) −4.28770 −0.401579
\(115\) −8.63066 −0.804813
\(116\) 7.92849 0.736142
\(117\) 0 0
\(118\) 3.95790 0.364354
\(119\) 2.69964 0.247476
\(120\) 23.0125 2.10075
\(121\) 24.3156 2.21051
\(122\) 0.952877 0.0862694
\(123\) 9.93284 0.895614
\(124\) −1.52207 −0.136686
\(125\) −0.00370455 −0.000331345 0
\(126\) 3.35776 0.299133
\(127\) 6.13117 0.544053 0.272027 0.962290i \(-0.412306\pi\)
0.272027 + 0.962290i \(0.412306\pi\)
\(128\) 8.30013 0.733635
\(129\) 18.5988 1.63753
\(130\) 0 0
\(131\) 10.2217 0.893073 0.446537 0.894765i \(-0.352657\pi\)
0.446537 + 0.894765i \(0.352657\pi\)
\(132\) −20.8960 −1.81877
\(133\) −1.95705 −0.169698
\(134\) −5.17126 −0.446729
\(135\) 9.06111 0.779856
\(136\) −7.38528 −0.633283
\(137\) 19.9475 1.70423 0.852116 0.523353i \(-0.175319\pi\)
0.852116 + 0.523353i \(0.175319\pi\)
\(138\) 5.97987 0.509041
\(139\) −20.3275 −1.72415 −0.862077 0.506777i \(-0.830837\pi\)
−0.862077 + 0.506777i \(0.830837\pi\)
\(140\) 4.17951 0.353233
\(141\) 1.21328 0.102177
\(142\) −3.71193 −0.311498
\(143\) 0 0
\(144\) 1.59233 0.132694
\(145\) −18.9677 −1.57518
\(146\) −6.84025 −0.566103
\(147\) 2.66029 0.219417
\(148\) 8.59741 0.706703
\(149\) −10.7162 −0.877901 −0.438951 0.898511i \(-0.644650\pi\)
−0.438951 + 0.898511i \(0.644650\pi\)
\(150\) −10.9519 −0.894221
\(151\) −8.74416 −0.711590 −0.355795 0.934564i \(-0.615790\pi\)
−0.355795 + 0.934564i \(0.615790\pi\)
\(152\) 5.35380 0.434251
\(153\) −11.0069 −0.889852
\(154\) 4.89414 0.394381
\(155\) 3.64130 0.292476
\(156\) 0 0
\(157\) 6.50734 0.519342 0.259671 0.965697i \(-0.416386\pi\)
0.259671 + 0.965697i \(0.416386\pi\)
\(158\) 6.51770 0.518520
\(159\) 1.06222 0.0842393
\(160\) −18.3178 −1.44815
\(161\) 2.72941 0.215108
\(162\) 3.79518 0.298177
\(163\) −2.61267 −0.204640 −0.102320 0.994752i \(-0.532627\pi\)
−0.102320 + 0.994752i \(0.532627\pi\)
\(164\) −4.93509 −0.385366
\(165\) 49.9905 3.89175
\(166\) 5.10436 0.396175
\(167\) −3.88624 −0.300726 −0.150363 0.988631i \(-0.548044\pi\)
−0.150363 + 0.988631i \(0.548044\pi\)
\(168\) −7.27763 −0.561482
\(169\) 0 0
\(170\) 7.03030 0.539200
\(171\) 7.97919 0.610184
\(172\) −9.24072 −0.704599
\(173\) 13.9768 1.06263 0.531317 0.847173i \(-0.321697\pi\)
0.531317 + 0.847173i \(0.321697\pi\)
\(174\) 13.1420 0.996293
\(175\) −4.99883 −0.377876
\(176\) 2.32091 0.174945
\(177\) −12.7850 −0.960979
\(178\) −2.93298 −0.219836
\(179\) −25.2843 −1.88984 −0.944919 0.327305i \(-0.893860\pi\)
−0.944919 + 0.327305i \(0.893860\pi\)
\(180\) −17.0405 −1.27013
\(181\) 0.864474 0.0642559 0.0321279 0.999484i \(-0.489772\pi\)
0.0321279 + 0.999484i \(0.489772\pi\)
\(182\) 0 0
\(183\) −3.07803 −0.227535
\(184\) −7.46673 −0.550454
\(185\) −20.5680 −1.51219
\(186\) −2.52293 −0.184990
\(187\) −16.0432 −1.17319
\(188\) −0.602814 −0.0439647
\(189\) −2.86554 −0.208438
\(190\) −5.09647 −0.369737
\(191\) 14.6676 1.06131 0.530657 0.847587i \(-0.321945\pi\)
0.530657 + 0.847587i \(0.321945\pi\)
\(192\) 10.6138 0.765985
\(193\) 16.4959 1.18740 0.593700 0.804686i \(-0.297667\pi\)
0.593700 + 0.804686i \(0.297667\pi\)
\(194\) 2.82275 0.202661
\(195\) 0 0
\(196\) −1.32176 −0.0944111
\(197\) −11.0102 −0.784443 −0.392222 0.919871i \(-0.628293\pi\)
−0.392222 + 0.919871i \(0.628293\pi\)
\(198\) −19.9542 −1.41808
\(199\) −21.2117 −1.50366 −0.751829 0.659358i \(-0.770828\pi\)
−0.751829 + 0.659358i \(0.770828\pi\)
\(200\) 13.6751 0.966972
\(201\) 16.7045 1.17824
\(202\) 10.9776 0.772380
\(203\) 5.99845 0.421009
\(204\) 9.49264 0.664617
\(205\) 11.8064 0.824597
\(206\) −9.59027 −0.668186
\(207\) −11.1282 −0.773466
\(208\) 0 0
\(209\) 11.6301 0.804474
\(210\) 6.92783 0.478065
\(211\) −17.9358 −1.23475 −0.617375 0.786669i \(-0.711804\pi\)
−0.617375 + 0.786669i \(0.711804\pi\)
\(212\) −0.527759 −0.0362466
\(213\) 11.9905 0.821572
\(214\) −3.23629 −0.221228
\(215\) 22.1070 1.50768
\(216\) 7.83913 0.533385
\(217\) −1.15155 −0.0781722
\(218\) 9.26774 0.627691
\(219\) 22.0957 1.49309
\(220\) −24.8376 −1.67455
\(221\) 0 0
\(222\) 14.2508 0.956451
\(223\) 16.0312 1.07353 0.536763 0.843733i \(-0.319647\pi\)
0.536763 + 0.843733i \(0.319647\pi\)
\(224\) 5.79294 0.387057
\(225\) 20.3810 1.35873
\(226\) 4.75536 0.316322
\(227\) −16.3750 −1.08685 −0.543424 0.839458i \(-0.682873\pi\)
−0.543424 + 0.839458i \(0.682873\pi\)
\(228\) −6.88148 −0.455737
\(229\) 27.0104 1.78490 0.892449 0.451148i \(-0.148985\pi\)
0.892449 + 0.451148i \(0.148985\pi\)
\(230\) 7.10783 0.468677
\(231\) −15.8093 −1.04018
\(232\) −16.4097 −1.07735
\(233\) 11.5681 0.757853 0.378926 0.925427i \(-0.376293\pi\)
0.378926 + 0.925427i \(0.376293\pi\)
\(234\) 0 0
\(235\) 1.44214 0.0940747
\(236\) 6.35217 0.413491
\(237\) −21.0538 −1.36759
\(238\) −2.22331 −0.144116
\(239\) 14.6731 0.949122 0.474561 0.880223i \(-0.342607\pi\)
0.474561 + 0.880223i \(0.342607\pi\)
\(240\) 3.28533 0.212067
\(241\) −14.3467 −0.924151 −0.462076 0.886841i \(-0.652895\pi\)
−0.462076 + 0.886841i \(0.652895\pi\)
\(242\) −20.0253 −1.28727
\(243\) −20.8560 −1.33791
\(244\) 1.52931 0.0979039
\(245\) 3.16209 0.202019
\(246\) −8.18025 −0.521554
\(247\) 0 0
\(248\) 3.15024 0.200040
\(249\) −16.4883 −1.04491
\(250\) 0.00305091 0.000192956 0
\(251\) 8.61452 0.543744 0.271872 0.962333i \(-0.412357\pi\)
0.271872 + 0.962333i \(0.412357\pi\)
\(252\) 5.38900 0.339475
\(253\) −16.2201 −1.01975
\(254\) −5.04936 −0.316825
\(255\) −22.7096 −1.42213
\(256\) −14.8151 −0.925941
\(257\) 10.3639 0.646485 0.323243 0.946316i \(-0.395227\pi\)
0.323243 + 0.946316i \(0.395227\pi\)
\(258\) −15.3171 −0.953603
\(259\) 6.50454 0.404172
\(260\) 0 0
\(261\) −24.4566 −1.51383
\(262\) −8.41813 −0.520074
\(263\) −22.0826 −1.36167 −0.680835 0.732436i \(-0.738383\pi\)
−0.680835 + 0.732436i \(0.738383\pi\)
\(264\) 43.2488 2.66178
\(265\) 1.26258 0.0775596
\(266\) 1.61174 0.0988220
\(267\) 9.47427 0.579816
\(268\) −8.29954 −0.506975
\(269\) 12.9399 0.788960 0.394480 0.918905i \(-0.370925\pi\)
0.394480 + 0.918905i \(0.370925\pi\)
\(270\) −7.46233 −0.454143
\(271\) −17.6749 −1.07367 −0.536837 0.843686i \(-0.680381\pi\)
−0.536837 + 0.843686i \(0.680381\pi\)
\(272\) −1.05434 −0.0639289
\(273\) 0 0
\(274\) −16.4279 −0.992446
\(275\) 29.7065 1.79137
\(276\) 9.59732 0.577691
\(277\) −18.0150 −1.08242 −0.541209 0.840888i \(-0.682033\pi\)
−0.541209 + 0.840888i \(0.682033\pi\)
\(278\) 16.7408 1.00405
\(279\) 4.69504 0.281085
\(280\) −8.65038 −0.516959
\(281\) 2.44178 0.145665 0.0728323 0.997344i \(-0.476796\pi\)
0.0728323 + 0.997344i \(0.476796\pi\)
\(282\) −0.999205 −0.0595018
\(283\) −28.7240 −1.70746 −0.853732 0.520713i \(-0.825666\pi\)
−0.853732 + 0.520713i \(0.825666\pi\)
\(284\) −5.95741 −0.353507
\(285\) 16.4629 0.975176
\(286\) 0 0
\(287\) −3.73374 −0.220396
\(288\) −23.6187 −1.39175
\(289\) −9.71193 −0.571290
\(290\) 15.6209 0.917292
\(291\) −9.11818 −0.534517
\(292\) −10.9782 −0.642449
\(293\) −29.3309 −1.71353 −0.856763 0.515710i \(-0.827528\pi\)
−0.856763 + 0.515710i \(0.827528\pi\)
\(294\) −2.19090 −0.127776
\(295\) −15.1966 −0.884779
\(296\) −17.7942 −1.03426
\(297\) 17.0291 0.988126
\(298\) 8.82535 0.511239
\(299\) 0 0
\(300\) −17.5772 −1.01482
\(301\) −6.99125 −0.402969
\(302\) 7.20131 0.414389
\(303\) −35.4603 −2.03714
\(304\) 0.764323 0.0438369
\(305\) −3.65863 −0.209492
\(306\) 9.06476 0.518198
\(307\) −7.06910 −0.403455 −0.201728 0.979442i \(-0.564656\pi\)
−0.201728 + 0.979442i \(0.564656\pi\)
\(308\) 7.85479 0.447568
\(309\) 30.9790 1.76233
\(310\) −2.99882 −0.170321
\(311\) −22.2686 −1.26274 −0.631368 0.775483i \(-0.717506\pi\)
−0.631368 + 0.775483i \(0.717506\pi\)
\(312\) 0 0
\(313\) −28.0840 −1.58740 −0.793700 0.608309i \(-0.791848\pi\)
−0.793700 + 0.608309i \(0.791848\pi\)
\(314\) −5.35916 −0.302435
\(315\) −12.8923 −0.726401
\(316\) 10.4605 0.588449
\(317\) 19.5155 1.09610 0.548049 0.836446i \(-0.315371\pi\)
0.548049 + 0.836446i \(0.315371\pi\)
\(318\) −0.874796 −0.0490561
\(319\) −35.6470 −1.99585
\(320\) 12.6158 0.705247
\(321\) 10.4540 0.583488
\(322\) −2.24783 −0.125266
\(323\) −5.28333 −0.293972
\(324\) 6.09102 0.338390
\(325\) 0 0
\(326\) 2.15168 0.119171
\(327\) −29.9371 −1.65553
\(328\) 10.2142 0.563986
\(329\) −0.456071 −0.0251440
\(330\) −41.1700 −2.26633
\(331\) 15.6308 0.859145 0.429573 0.903032i \(-0.358664\pi\)
0.429573 + 0.903032i \(0.358664\pi\)
\(332\) 8.19217 0.449604
\(333\) −26.5200 −1.45329
\(334\) 3.20053 0.175125
\(335\) 19.8553 1.08481
\(336\) −1.03897 −0.0566807
\(337\) −21.7501 −1.18480 −0.592401 0.805643i \(-0.701820\pi\)
−0.592401 + 0.805643i \(0.701820\pi\)
\(338\) 0 0
\(339\) −15.3610 −0.834295
\(340\) 11.2832 0.611917
\(341\) 6.84330 0.370586
\(342\) −6.57131 −0.355336
\(343\) −1.00000 −0.0539949
\(344\) 19.1256 1.03118
\(345\) −22.9601 −1.23613
\(346\) −11.5106 −0.618816
\(347\) −15.9590 −0.856726 −0.428363 0.903607i \(-0.640909\pi\)
−0.428363 + 0.903607i \(0.640909\pi\)
\(348\) 21.0921 1.13065
\(349\) 6.81706 0.364909 0.182455 0.983214i \(-0.441596\pi\)
0.182455 + 0.983214i \(0.441596\pi\)
\(350\) 4.11682 0.220053
\(351\) 0 0
\(352\) −34.4257 −1.83490
\(353\) −14.0033 −0.745318 −0.372659 0.927968i \(-0.621554\pi\)
−0.372659 + 0.927968i \(0.621554\pi\)
\(354\) 10.5292 0.559619
\(355\) 14.2522 0.756427
\(356\) −4.70725 −0.249484
\(357\) 7.18184 0.380103
\(358\) 20.8230 1.10053
\(359\) 5.41494 0.285789 0.142895 0.989738i \(-0.454359\pi\)
0.142895 + 0.989738i \(0.454359\pi\)
\(360\) 35.2689 1.85884
\(361\) −15.1700 −0.798419
\(362\) −0.711943 −0.0374189
\(363\) 64.6867 3.39517
\(364\) 0 0
\(365\) 26.2636 1.37470
\(366\) 2.53493 0.132503
\(367\) 30.0317 1.56764 0.783822 0.620985i \(-0.213267\pi\)
0.783822 + 0.620985i \(0.213267\pi\)
\(368\) −1.06597 −0.0555675
\(369\) 15.2230 0.792480
\(370\) 16.9389 0.880610
\(371\) −0.399286 −0.0207299
\(372\) −4.04914 −0.209938
\(373\) 21.4098 1.10856 0.554278 0.832332i \(-0.312995\pi\)
0.554278 + 0.832332i \(0.312995\pi\)
\(374\) 13.2124 0.683199
\(375\) −0.00985519 −0.000508920 0
\(376\) 1.24765 0.0643427
\(377\) 0 0
\(378\) 2.35994 0.121382
\(379\) 9.47655 0.486778 0.243389 0.969929i \(-0.421741\pi\)
0.243389 + 0.969929i \(0.421741\pi\)
\(380\) −8.17951 −0.419600
\(381\) 16.3107 0.835622
\(382\) −12.0796 −0.618048
\(383\) −5.43061 −0.277491 −0.138746 0.990328i \(-0.544307\pi\)
−0.138746 + 0.990328i \(0.544307\pi\)
\(384\) 22.0808 1.12680
\(385\) −18.7914 −0.957696
\(386\) −13.5853 −0.691473
\(387\) 28.5044 1.44896
\(388\) 4.53033 0.229993
\(389\) −10.6422 −0.539580 −0.269790 0.962919i \(-0.586954\pi\)
−0.269790 + 0.962919i \(0.586954\pi\)
\(390\) 0 0
\(391\) 7.36845 0.372638
\(392\) 2.73565 0.138171
\(393\) 27.1927 1.37169
\(394\) 9.06750 0.456814
\(395\) −25.0251 −1.25915
\(396\) −32.0252 −1.60933
\(397\) −37.1854 −1.86628 −0.933140 0.359512i \(-0.882943\pi\)
−0.933140 + 0.359512i \(0.882943\pi\)
\(398\) 17.4690 0.875644
\(399\) −5.20632 −0.260642
\(400\) 1.95229 0.0976143
\(401\) −0.896610 −0.0447746 −0.0223873 0.999749i \(-0.507127\pi\)
−0.0223873 + 0.999749i \(0.507127\pi\)
\(402\) −13.7571 −0.686139
\(403\) 0 0
\(404\) 17.6183 0.876545
\(405\) −14.5718 −0.724079
\(406\) −4.94006 −0.245171
\(407\) −38.6545 −1.91603
\(408\) −19.6470 −0.972672
\(409\) 24.5773 1.21527 0.607635 0.794217i \(-0.292119\pi\)
0.607635 + 0.794217i \(0.292119\pi\)
\(410\) −9.72326 −0.480198
\(411\) 53.0662 2.61756
\(412\) −15.3918 −0.758299
\(413\) 4.80586 0.236481
\(414\) 9.16473 0.450422
\(415\) −19.5985 −0.962052
\(416\) 0 0
\(417\) −54.0770 −2.64816
\(418\) −9.57807 −0.468479
\(419\) −7.64558 −0.373511 −0.186755 0.982406i \(-0.559797\pi\)
−0.186755 + 0.982406i \(0.559797\pi\)
\(420\) 11.1187 0.542538
\(421\) −25.0780 −1.22223 −0.611113 0.791544i \(-0.709278\pi\)
−0.611113 + 0.791544i \(0.709278\pi\)
\(422\) 14.7711 0.719046
\(423\) 1.85947 0.0904106
\(424\) 1.09231 0.0530471
\(425\) −13.4951 −0.654606
\(426\) −9.87481 −0.478436
\(427\) 1.15703 0.0559925
\(428\) −5.19405 −0.251064
\(429\) 0 0
\(430\) −18.2063 −0.877987
\(431\) −7.75404 −0.373499 −0.186750 0.982408i \(-0.559795\pi\)
−0.186750 + 0.982408i \(0.559795\pi\)
\(432\) 1.11913 0.0538444
\(433\) 35.9760 1.72890 0.864448 0.502722i \(-0.167668\pi\)
0.864448 + 0.502722i \(0.167668\pi\)
\(434\) 0.948365 0.0455230
\(435\) −50.4595 −2.41935
\(436\) 14.8741 0.712342
\(437\) −5.34160 −0.255523
\(438\) −18.1971 −0.869489
\(439\) −28.2350 −1.34758 −0.673792 0.738921i \(-0.735336\pi\)
−0.673792 + 0.738921i \(0.735336\pi\)
\(440\) 51.4066 2.45071
\(441\) 4.07715 0.194150
\(442\) 0 0
\(443\) 28.7918 1.36794 0.683970 0.729511i \(-0.260252\pi\)
0.683970 + 0.729511i \(0.260252\pi\)
\(444\) 22.8716 1.08544
\(445\) 11.2614 0.533840
\(446\) −13.2026 −0.625159
\(447\) −28.5081 −1.34839
\(448\) −3.98971 −0.188496
\(449\) 29.1902 1.37757 0.688785 0.724965i \(-0.258144\pi\)
0.688785 + 0.724965i \(0.258144\pi\)
\(450\) −16.7849 −0.791247
\(451\) 22.1885 1.04482
\(452\) 7.63205 0.358982
\(453\) −23.2620 −1.09295
\(454\) 13.4858 0.632918
\(455\) 0 0
\(456\) 14.2427 0.666974
\(457\) 31.6848 1.48215 0.741077 0.671420i \(-0.234316\pi\)
0.741077 + 0.671420i \(0.234316\pi\)
\(458\) −22.2446 −1.03942
\(459\) −7.73594 −0.361083
\(460\) 11.4076 0.531883
\(461\) −22.1018 −1.02938 −0.514691 0.857376i \(-0.672093\pi\)
−0.514691 + 0.857376i \(0.672093\pi\)
\(462\) 13.0199 0.605738
\(463\) −38.8811 −1.80696 −0.903479 0.428632i \(-0.858996\pi\)
−0.903479 + 0.428632i \(0.858996\pi\)
\(464\) −2.34269 −0.108757
\(465\) 9.68693 0.449220
\(466\) −9.52699 −0.441329
\(467\) 13.2823 0.614632 0.307316 0.951607i \(-0.400569\pi\)
0.307316 + 0.951607i \(0.400569\pi\)
\(468\) 0 0
\(469\) −6.27918 −0.289946
\(470\) −1.18768 −0.0547837
\(471\) 17.3114 0.797668
\(472\) −13.1472 −0.605147
\(473\) 41.5469 1.91033
\(474\) 17.3390 0.796406
\(475\) 9.78295 0.448872
\(476\) −3.56827 −0.163551
\(477\) 1.62795 0.0745387
\(478\) −12.0841 −0.552714
\(479\) 6.63512 0.303166 0.151583 0.988445i \(-0.451563\pi\)
0.151583 + 0.988445i \(0.451563\pi\)
\(480\) −48.7307 −2.22424
\(481\) 0 0
\(482\) 11.8153 0.538172
\(483\) 7.26104 0.330389
\(484\) −32.1393 −1.46088
\(485\) −10.8381 −0.492133
\(486\) 17.1761 0.779123
\(487\) 33.4701 1.51668 0.758338 0.651861i \(-0.226012\pi\)
0.758338 + 0.651861i \(0.226012\pi\)
\(488\) −3.16522 −0.143283
\(489\) −6.95047 −0.314311
\(490\) −2.60416 −0.117644
\(491\) −37.3287 −1.68462 −0.842310 0.538993i \(-0.818805\pi\)
−0.842310 + 0.538993i \(0.818805\pi\)
\(492\) −13.1288 −0.591891
\(493\) 16.1937 0.729327
\(494\) 0 0
\(495\) 76.6152 3.44360
\(496\) 0.449736 0.0201937
\(497\) −4.50720 −0.202175
\(498\) 13.5791 0.608493
\(499\) 34.1327 1.52799 0.763994 0.645223i \(-0.223236\pi\)
0.763994 + 0.645223i \(0.223236\pi\)
\(500\) 0.00489651 0.000218979 0
\(501\) −10.3385 −0.461891
\(502\) −7.09454 −0.316645
\(503\) 15.3089 0.682592 0.341296 0.939956i \(-0.389134\pi\)
0.341296 + 0.939956i \(0.389134\pi\)
\(504\) −11.1537 −0.496824
\(505\) −42.1491 −1.87561
\(506\) 13.3581 0.593842
\(507\) 0 0
\(508\) −8.10390 −0.359553
\(509\) 18.4970 0.819866 0.409933 0.912116i \(-0.365552\pi\)
0.409933 + 0.912116i \(0.365552\pi\)
\(510\) 18.7027 0.828168
\(511\) −8.30575 −0.367425
\(512\) −4.39924 −0.194421
\(513\) 5.60801 0.247599
\(514\) −8.53529 −0.376476
\(515\) 36.8224 1.62259
\(516\) −24.5830 −1.08221
\(517\) 2.71029 0.119198
\(518\) −5.35685 −0.235367
\(519\) 37.1823 1.63212
\(520\) 0 0
\(521\) 23.5865 1.03334 0.516671 0.856184i \(-0.327171\pi\)
0.516671 + 0.856184i \(0.327171\pi\)
\(522\) 20.1414 0.881564
\(523\) 12.3059 0.538099 0.269049 0.963126i \(-0.413291\pi\)
0.269049 + 0.963126i \(0.413291\pi\)
\(524\) −13.5106 −0.590212
\(525\) −13.2983 −0.580387
\(526\) 18.1862 0.792958
\(527\) −3.10877 −0.135420
\(528\) 6.17431 0.268702
\(529\) −15.5503 −0.676100
\(530\) −1.03980 −0.0451662
\(531\) −19.5942 −0.850317
\(532\) 2.58674 0.112149
\(533\) 0 0
\(534\) −7.80259 −0.337651
\(535\) 12.4259 0.537220
\(536\) 17.1777 0.741962
\(537\) −67.2636 −2.90264
\(538\) −10.6567 −0.459444
\(539\) 5.94270 0.255970
\(540\) −11.9766 −0.515390
\(541\) 19.4411 0.835838 0.417919 0.908484i \(-0.362760\pi\)
0.417919 + 0.908484i \(0.362760\pi\)
\(542\) 14.5563 0.625245
\(543\) 2.29975 0.0986919
\(544\) 15.6389 0.670511
\(545\) −35.5841 −1.52425
\(546\) 0 0
\(547\) 40.2163 1.71953 0.859763 0.510693i \(-0.170611\pi\)
0.859763 + 0.510693i \(0.170611\pi\)
\(548\) −26.3658 −1.12629
\(549\) −4.71738 −0.201333
\(550\) −24.4650 −1.04319
\(551\) −11.7393 −0.500109
\(552\) −19.8637 −0.845454
\(553\) 7.91410 0.336542
\(554\) 14.8364 0.630337
\(555\) −54.7168 −2.32260
\(556\) 26.8680 1.13945
\(557\) 7.96399 0.337445 0.168722 0.985664i \(-0.446036\pi\)
0.168722 + 0.985664i \(0.446036\pi\)
\(558\) −3.86663 −0.163687
\(559\) 0 0
\(560\) −1.23495 −0.0521862
\(561\) −42.6795 −1.80193
\(562\) −2.01095 −0.0848266
\(563\) −1.42396 −0.0600128 −0.0300064 0.999550i \(-0.509553\pi\)
−0.0300064 + 0.999550i \(0.509553\pi\)
\(564\) −1.60366 −0.0675263
\(565\) −18.2585 −0.768140
\(566\) 23.6558 0.994327
\(567\) 4.60828 0.193530
\(568\) 12.3301 0.517360
\(569\) −18.5189 −0.776353 −0.388177 0.921585i \(-0.626895\pi\)
−0.388177 + 0.921585i \(0.626895\pi\)
\(570\) −13.5581 −0.567886
\(571\) 4.35766 0.182362 0.0911812 0.995834i \(-0.470936\pi\)
0.0911812 + 0.995834i \(0.470936\pi\)
\(572\) 0 0
\(573\) 39.0202 1.63009
\(574\) 3.07494 0.128346
\(575\) −13.6439 −0.568989
\(576\) 16.2667 0.677778
\(577\) −9.56416 −0.398161 −0.199081 0.979983i \(-0.563796\pi\)
−0.199081 + 0.979983i \(0.563796\pi\)
\(578\) 7.99832 0.332686
\(579\) 43.8839 1.82375
\(580\) 25.0706 1.04100
\(581\) 6.19795 0.257134
\(582\) 7.50933 0.311272
\(583\) 2.37284 0.0982728
\(584\) 22.7216 0.940228
\(585\) 0 0
\(586\) 24.1556 0.997859
\(587\) 2.36053 0.0974296 0.0487148 0.998813i \(-0.484487\pi\)
0.0487148 + 0.998813i \(0.484487\pi\)
\(588\) −3.51626 −0.145008
\(589\) 2.25364 0.0928594
\(590\) 12.5152 0.515244
\(591\) −29.2903 −1.20484
\(592\) −2.54034 −0.104407
\(593\) 40.4292 1.66023 0.830114 0.557594i \(-0.188275\pi\)
0.830114 + 0.557594i \(0.188275\pi\)
\(594\) −14.0244 −0.575427
\(595\) 8.53652 0.349963
\(596\) 14.1641 0.580185
\(597\) −56.4294 −2.30950
\(598\) 0 0
\(599\) −38.5873 −1.57663 −0.788316 0.615270i \(-0.789047\pi\)
−0.788316 + 0.615270i \(0.789047\pi\)
\(600\) 36.3796 1.48519
\(601\) 8.16231 0.332948 0.166474 0.986046i \(-0.446762\pi\)
0.166474 + 0.986046i \(0.446762\pi\)
\(602\) 5.75769 0.234666
\(603\) 25.6012 1.04256
\(604\) 11.5576 0.470274
\(605\) 76.8883 3.12595
\(606\) 29.2036 1.18631
\(607\) 7.58525 0.307876 0.153938 0.988081i \(-0.450804\pi\)
0.153938 + 0.988081i \(0.450804\pi\)
\(608\) −11.3371 −0.459779
\(609\) 15.9576 0.646636
\(610\) 3.01308 0.121996
\(611\) 0 0
\(612\) 14.5484 0.588083
\(613\) 15.5778 0.629183 0.314592 0.949227i \(-0.398132\pi\)
0.314592 + 0.949227i \(0.398132\pi\)
\(614\) 5.82180 0.234949
\(615\) 31.4086 1.26652
\(616\) −16.2571 −0.655019
\(617\) 23.8687 0.960916 0.480458 0.877018i \(-0.340471\pi\)
0.480458 + 0.877018i \(0.340471\pi\)
\(618\) −25.5129 −1.02628
\(619\) −19.4963 −0.783622 −0.391811 0.920046i \(-0.628151\pi\)
−0.391811 + 0.920046i \(0.628151\pi\)
\(620\) −4.81291 −0.193291
\(621\) −7.82126 −0.313856
\(622\) 18.3394 0.735344
\(623\) −3.56136 −0.142683
\(624\) 0 0
\(625\) −25.0059 −1.00023
\(626\) 23.1287 0.924410
\(627\) 30.9396 1.23561
\(628\) −8.60111 −0.343222
\(629\) 17.5599 0.700161
\(630\) 10.6176 0.423014
\(631\) 25.6619 1.02158 0.510792 0.859704i \(-0.329352\pi\)
0.510792 + 0.859704i \(0.329352\pi\)
\(632\) −21.6502 −0.861199
\(633\) −47.7144 −1.89648
\(634\) −16.0721 −0.638304
\(635\) 19.3873 0.769362
\(636\) −1.40399 −0.0556719
\(637\) 0 0
\(638\) 29.3573 1.16227
\(639\) 18.3765 0.726964
\(640\) 26.2458 1.03746
\(641\) −1.10604 −0.0436860 −0.0218430 0.999761i \(-0.506953\pi\)
−0.0218430 + 0.999761i \(0.506953\pi\)
\(642\) −8.60949 −0.339789
\(643\) 12.6367 0.498341 0.249171 0.968460i \(-0.419842\pi\)
0.249171 + 0.968460i \(0.419842\pi\)
\(644\) −3.60762 −0.142160
\(645\) 58.8110 2.31568
\(646\) 4.35112 0.171192
\(647\) 25.7148 1.01095 0.505477 0.862840i \(-0.331317\pi\)
0.505477 + 0.862840i \(0.331317\pi\)
\(648\) −12.6066 −0.495236
\(649\) −28.5598 −1.12107
\(650\) 0 0
\(651\) −3.06345 −0.120066
\(652\) 3.45331 0.135242
\(653\) 25.2607 0.988527 0.494263 0.869312i \(-0.335438\pi\)
0.494263 + 0.869312i \(0.335438\pi\)
\(654\) 24.6549 0.964083
\(655\) 32.3219 1.26292
\(656\) 1.45821 0.0569335
\(657\) 33.8638 1.32115
\(658\) 0.375600 0.0146424
\(659\) 22.9764 0.895034 0.447517 0.894275i \(-0.352308\pi\)
0.447517 + 0.894275i \(0.352308\pi\)
\(660\) −66.0752 −2.57197
\(661\) 30.4326 1.18369 0.591845 0.806051i \(-0.298400\pi\)
0.591845 + 0.806051i \(0.298400\pi\)
\(662\) −12.8728 −0.500316
\(663\) 0 0
\(664\) −16.9554 −0.657998
\(665\) −6.18837 −0.239975
\(666\) 21.8407 0.846310
\(667\) 16.3723 0.633937
\(668\) 5.13665 0.198743
\(669\) 42.6475 1.64885
\(670\) −16.3520 −0.631733
\(671\) −6.87586 −0.265440
\(672\) 15.4109 0.594489
\(673\) 10.8387 0.417800 0.208900 0.977937i \(-0.433012\pi\)
0.208900 + 0.977937i \(0.433012\pi\)
\(674\) 17.9124 0.689960
\(675\) 14.3244 0.551345
\(676\) 0 0
\(677\) −18.1209 −0.696442 −0.348221 0.937412i \(-0.613214\pi\)
−0.348221 + 0.937412i \(0.613214\pi\)
\(678\) 12.6506 0.485845
\(679\) 3.42751 0.131536
\(680\) −23.3529 −0.895545
\(681\) −43.5624 −1.66931
\(682\) −5.63584 −0.215808
\(683\) −37.8352 −1.44772 −0.723861 0.689946i \(-0.757634\pi\)
−0.723861 + 0.689946i \(0.757634\pi\)
\(684\) −10.5465 −0.403257
\(685\) 63.0759 2.41001
\(686\) 0.823556 0.0314435
\(687\) 71.8556 2.74146
\(688\) 2.73042 0.104096
\(689\) 0 0
\(690\) 18.9089 0.719850
\(691\) −30.0261 −1.14225 −0.571124 0.820864i \(-0.693492\pi\)
−0.571124 + 0.820864i \(0.693492\pi\)
\(692\) −18.4739 −0.702271
\(693\) −24.2293 −0.920394
\(694\) 13.1432 0.498907
\(695\) −64.2774 −2.43818
\(696\) −43.6545 −1.65472
\(697\) −10.0798 −0.381798
\(698\) −5.61423 −0.212502
\(699\) 30.7746 1.16400
\(700\) 6.60723 0.249730
\(701\) −0.116177 −0.00438796 −0.00219398 0.999998i \(-0.500698\pi\)
−0.00219398 + 0.999998i \(0.500698\pi\)
\(702\) 0 0
\(703\) −12.7297 −0.480110
\(704\) 23.7097 0.893591
\(705\) 3.83651 0.144491
\(706\) 11.5325 0.434030
\(707\) 13.3295 0.501307
\(708\) 16.8986 0.635090
\(709\) −6.72993 −0.252748 −0.126374 0.991983i \(-0.540334\pi\)
−0.126374 + 0.991983i \(0.540334\pi\)
\(710\) −11.7375 −0.440499
\(711\) −32.2670 −1.21011
\(712\) 9.74265 0.365121
\(713\) −3.14305 −0.117708
\(714\) −5.91465 −0.221350
\(715\) 0 0
\(716\) 33.4197 1.24895
\(717\) 39.0347 1.45778
\(718\) −4.45950 −0.166427
\(719\) 46.8078 1.74564 0.872818 0.488046i \(-0.162290\pi\)
0.872818 + 0.488046i \(0.162290\pi\)
\(720\) 5.03509 0.187647
\(721\) −11.6450 −0.433681
\(722\) 12.4933 0.464953
\(723\) −38.1664 −1.41942
\(724\) −1.14262 −0.0424653
\(725\) −29.9852 −1.11362
\(726\) −53.2731 −1.97715
\(727\) −13.3362 −0.494611 −0.247305 0.968938i \(-0.579545\pi\)
−0.247305 + 0.968938i \(0.579545\pi\)
\(728\) 0 0
\(729\) −41.6582 −1.54290
\(730\) −21.6295 −0.800544
\(731\) −18.8739 −0.698076
\(732\) 4.06840 0.150373
\(733\) −29.4612 −1.08817 −0.544087 0.839029i \(-0.683124\pi\)
−0.544087 + 0.839029i \(0.683124\pi\)
\(734\) −24.7328 −0.912905
\(735\) 8.41209 0.310285
\(736\) 15.8113 0.582814
\(737\) 37.3153 1.37453
\(738\) −12.5370 −0.461494
\(739\) −12.0302 −0.442537 −0.221269 0.975213i \(-0.571020\pi\)
−0.221269 + 0.975213i \(0.571020\pi\)
\(740\) 27.1858 0.999370
\(741\) 0 0
\(742\) 0.328834 0.0120719
\(743\) 21.8826 0.802796 0.401398 0.915904i \(-0.368524\pi\)
0.401398 + 0.915904i \(0.368524\pi\)
\(744\) 8.38055 0.307246
\(745\) −33.8855 −1.24147
\(746\) −17.6321 −0.645558
\(747\) −25.2700 −0.924580
\(748\) 21.2051 0.775337
\(749\) −3.92966 −0.143587
\(750\) 0.00811630 0.000296365 0
\(751\) 34.7492 1.26802 0.634008 0.773327i \(-0.281409\pi\)
0.634008 + 0.773327i \(0.281409\pi\)
\(752\) 0.178118 0.00649529
\(753\) 22.9172 0.835147
\(754\) 0 0
\(755\) −27.6498 −1.00628
\(756\) 3.78755 0.137752
\(757\) 43.9263 1.59653 0.798265 0.602307i \(-0.205752\pi\)
0.798265 + 0.602307i \(0.205752\pi\)
\(758\) −7.80447 −0.283471
\(759\) −43.1502 −1.56625
\(760\) 16.9292 0.614087
\(761\) −0.141391 −0.00512543 −0.00256272 0.999997i \(-0.500816\pi\)
−0.00256272 + 0.999997i \(0.500816\pi\)
\(762\) −13.4328 −0.486618
\(763\) 11.2533 0.407398
\(764\) −19.3870 −0.701399
\(765\) −34.8047 −1.25837
\(766\) 4.47241 0.161595
\(767\) 0 0
\(768\) −39.4124 −1.42217
\(769\) −13.6486 −0.492180 −0.246090 0.969247i \(-0.579146\pi\)
−0.246090 + 0.969247i \(0.579146\pi\)
\(770\) 15.4757 0.557707
\(771\) 27.5711 0.992950
\(772\) −21.8035 −0.784726
\(773\) 17.5894 0.632646 0.316323 0.948652i \(-0.397552\pi\)
0.316323 + 0.948652i \(0.397552\pi\)
\(774\) −23.4750 −0.843790
\(775\) 5.75639 0.206776
\(776\) −9.37647 −0.336596
\(777\) 17.3040 0.620777
\(778\) 8.76443 0.314220
\(779\) 7.30711 0.261804
\(780\) 0 0
\(781\) 26.7849 0.958439
\(782\) −6.06833 −0.217003
\(783\) −17.1888 −0.614278
\(784\) 0.390549 0.0139482
\(785\) 20.5768 0.734418
\(786\) −22.3947 −0.798792
\(787\) −2.96845 −0.105814 −0.0529069 0.998599i \(-0.516849\pi\)
−0.0529069 + 0.998599i \(0.516849\pi\)
\(788\) 14.5528 0.518421
\(789\) −58.7461 −2.09142
\(790\) 20.6096 0.733256
\(791\) 5.77418 0.205306
\(792\) 66.2829 2.35526
\(793\) 0 0
\(794\) 30.6242 1.08681
\(795\) 3.35883 0.119125
\(796\) 28.0367 0.993735
\(797\) 9.45221 0.334815 0.167407 0.985888i \(-0.446461\pi\)
0.167407 + 0.985888i \(0.446461\pi\)
\(798\) 4.28770 0.151783
\(799\) −1.23123 −0.0435577
\(800\) −28.9579 −1.02382
\(801\) 14.5202 0.513047
\(802\) 0.738409 0.0260741
\(803\) 49.3586 1.74183
\(804\) −22.0792 −0.778674
\(805\) 8.63066 0.304191
\(806\) 0 0
\(807\) 34.4239 1.21178
\(808\) −36.4648 −1.28283
\(809\) −1.16255 −0.0408729 −0.0204365 0.999791i \(-0.506506\pi\)
−0.0204365 + 0.999791i \(0.506506\pi\)
\(810\) 12.0007 0.421662
\(811\) −19.5561 −0.686706 −0.343353 0.939206i \(-0.611563\pi\)
−0.343353 + 0.939206i \(0.611563\pi\)
\(812\) −7.92849 −0.278235
\(813\) −47.0204 −1.64908
\(814\) 31.8342 1.11579
\(815\) −8.26151 −0.289388
\(816\) −2.80486 −0.0981897
\(817\) 13.6822 0.478680
\(818\) −20.2408 −0.707702
\(819\) 0 0
\(820\) −15.6052 −0.544958
\(821\) −12.6189 −0.440403 −0.220201 0.975454i \(-0.570671\pi\)
−0.220201 + 0.975454i \(0.570671\pi\)
\(822\) −43.7030 −1.52432
\(823\) −6.56808 −0.228949 −0.114474 0.993426i \(-0.536518\pi\)
−0.114474 + 0.993426i \(0.536518\pi\)
\(824\) 31.8565 1.10978
\(825\) 79.0280 2.75140
\(826\) −3.95790 −0.137713
\(827\) −17.3050 −0.601754 −0.300877 0.953663i \(-0.597279\pi\)
−0.300877 + 0.953663i \(0.597279\pi\)
\(828\) 14.7088 0.511167
\(829\) −3.75674 −0.130477 −0.0652385 0.997870i \(-0.520781\pi\)
−0.0652385 + 0.997870i \(0.520781\pi\)
\(830\) 16.1404 0.560243
\(831\) −47.9252 −1.66251
\(832\) 0 0
\(833\) −2.69964 −0.0935371
\(834\) 44.5355 1.54214
\(835\) −12.2886 −0.425266
\(836\) −15.3722 −0.531659
\(837\) 3.29981 0.114058
\(838\) 6.29656 0.217511
\(839\) 46.3427 1.59993 0.799965 0.600047i \(-0.204852\pi\)
0.799965 + 0.600047i \(0.204852\pi\)
\(840\) −23.0125 −0.794008
\(841\) 6.98142 0.240739
\(842\) 20.6531 0.711753
\(843\) 6.49586 0.223729
\(844\) 23.7067 0.816018
\(845\) 0 0
\(846\) −1.53138 −0.0526499
\(847\) −24.3156 −0.835495
\(848\) 0.155941 0.00535503
\(849\) −76.4142 −2.62253
\(850\) 11.1139 0.381205
\(851\) 17.7536 0.608585
\(852\) −15.8485 −0.542959
\(853\) −15.3103 −0.524215 −0.262107 0.965039i \(-0.584417\pi\)
−0.262107 + 0.965039i \(0.584417\pi\)
\(854\) −0.952877 −0.0326068
\(855\) 25.2309 0.862879
\(856\) 10.7502 0.367433
\(857\) 2.59248 0.0885574 0.0442787 0.999019i \(-0.485901\pi\)
0.0442787 + 0.999019i \(0.485901\pi\)
\(858\) 0 0
\(859\) −13.7738 −0.469955 −0.234978 0.972001i \(-0.575502\pi\)
−0.234978 + 0.972001i \(0.575502\pi\)
\(860\) −29.2200 −0.996394
\(861\) −9.93284 −0.338510
\(862\) 6.38589 0.217504
\(863\) −29.7592 −1.01302 −0.506508 0.862235i \(-0.669064\pi\)
−0.506508 + 0.862235i \(0.669064\pi\)
\(864\) −16.5999 −0.564741
\(865\) 44.1958 1.50270
\(866\) −29.6282 −1.00681
\(867\) −25.8366 −0.877456
\(868\) 1.52207 0.0516623
\(869\) −47.0311 −1.59542
\(870\) 41.5562 1.40889
\(871\) 0 0
\(872\) −30.7852 −1.04252
\(873\) −13.9745 −0.472965
\(874\) 4.39910 0.148802
\(875\) 0.00370455 0.000125237 0
\(876\) −29.2051 −0.986750
\(877\) −1.44332 −0.0487374 −0.0243687 0.999703i \(-0.507758\pi\)
−0.0243687 + 0.999703i \(0.507758\pi\)
\(878\) 23.2531 0.784755
\(879\) −78.0286 −2.63184
\(880\) 7.33894 0.247396
\(881\) −35.8804 −1.20884 −0.604420 0.796666i \(-0.706595\pi\)
−0.604420 + 0.796666i \(0.706595\pi\)
\(882\) −3.35776 −0.113062
\(883\) −10.5626 −0.355458 −0.177729 0.984079i \(-0.556875\pi\)
−0.177729 + 0.984079i \(0.556875\pi\)
\(884\) 0 0
\(885\) −40.4273 −1.35895
\(886\) −23.7116 −0.796608
\(887\) 12.2280 0.410577 0.205288 0.978702i \(-0.434187\pi\)
0.205288 + 0.978702i \(0.434187\pi\)
\(888\) −47.3377 −1.58855
\(889\) −6.13117 −0.205633
\(890\) −9.27436 −0.310877
\(891\) −27.3856 −0.917452
\(892\) −21.1893 −0.709469
\(893\) 0.892553 0.0298681
\(894\) 23.4780 0.785222
\(895\) −79.9513 −2.67248
\(896\) −8.30013 −0.277288
\(897\) 0 0
\(898\) −24.0398 −0.802217
\(899\) −6.90751 −0.230378
\(900\) −26.9387 −0.897956
\(901\) −1.07793 −0.0359110
\(902\) −18.2735 −0.608440
\(903\) −18.5988 −0.618928
\(904\) −15.7961 −0.525372
\(905\) 2.73355 0.0908662
\(906\) 19.1576 0.636468
\(907\) 4.52555 0.150269 0.0751343 0.997173i \(-0.476061\pi\)
0.0751343 + 0.997173i \(0.476061\pi\)
\(908\) 21.6438 0.718274
\(909\) −54.3464 −1.80256
\(910\) 0 0
\(911\) −57.2723 −1.89751 −0.948757 0.316006i \(-0.897658\pi\)
−0.948757 + 0.316006i \(0.897658\pi\)
\(912\) 2.03332 0.0673300
\(913\) −36.8325 −1.21898
\(914\) −26.0942 −0.863120
\(915\) −9.73302 −0.321764
\(916\) −35.7012 −1.17960
\(917\) −10.2217 −0.337550
\(918\) 6.37098 0.210274
\(919\) −40.7551 −1.34439 −0.672193 0.740376i \(-0.734648\pi\)
−0.672193 + 0.740376i \(0.734648\pi\)
\(920\) −23.6105 −0.778415
\(921\) −18.8059 −0.619675
\(922\) 18.2021 0.599453
\(923\) 0 0
\(924\) 20.8960 0.687429
\(925\) −32.5151 −1.06909
\(926\) 32.0208 1.05227
\(927\) 47.4783 1.55939
\(928\) 34.7487 1.14068
\(929\) 52.2791 1.71522 0.857611 0.514298i \(-0.171948\pi\)
0.857611 + 0.514298i \(0.171948\pi\)
\(930\) −7.97773 −0.261600
\(931\) 1.95705 0.0641397
\(932\) −15.2902 −0.500848
\(933\) −59.2410 −1.93946
\(934\) −10.9387 −0.357926
\(935\) −50.7300 −1.65905
\(936\) 0 0
\(937\) −6.38634 −0.208633 −0.104316 0.994544i \(-0.533265\pi\)
−0.104316 + 0.994544i \(0.533265\pi\)
\(938\) 5.17126 0.168848
\(939\) −74.7116 −2.43812
\(940\) −1.90615 −0.0621719
\(941\) 25.3711 0.827073 0.413536 0.910488i \(-0.364293\pi\)
0.413536 + 0.910488i \(0.364293\pi\)
\(942\) −14.2569 −0.464516
\(943\) −10.1909 −0.331862
\(944\) −1.87692 −0.0610887
\(945\) −9.06111 −0.294758
\(946\) −34.2162 −1.11246
\(947\) 27.2061 0.884080 0.442040 0.896995i \(-0.354255\pi\)
0.442040 + 0.896995i \(0.354255\pi\)
\(948\) 27.8280 0.903811
\(949\) 0 0
\(950\) −8.05681 −0.261397
\(951\) 51.9169 1.68352
\(952\) 7.38528 0.239358
\(953\) 26.5879 0.861265 0.430633 0.902527i \(-0.358290\pi\)
0.430633 + 0.902527i \(0.358290\pi\)
\(954\) −1.34071 −0.0434070
\(955\) 46.3805 1.50084
\(956\) −19.3942 −0.627254
\(957\) −94.8314 −3.06546
\(958\) −5.46439 −0.176546
\(959\) −19.9475 −0.644139
\(960\) 33.5618 1.08320
\(961\) −29.6739 −0.957224
\(962\) 0 0
\(963\) 16.0218 0.516296
\(964\) 18.9628 0.610751
\(965\) 52.1615 1.67914
\(966\) −5.97987 −0.192399
\(967\) −35.2467 −1.13346 −0.566729 0.823904i \(-0.691791\pi\)
−0.566729 + 0.823904i \(0.691791\pi\)
\(968\) 66.5191 2.13801
\(969\) −14.0552 −0.451518
\(970\) 8.92578 0.286590
\(971\) 36.9783 1.18669 0.593344 0.804949i \(-0.297807\pi\)
0.593344 + 0.804949i \(0.297807\pi\)
\(972\) 27.5665 0.884197
\(973\) 20.3275 0.651669
\(974\) −27.5645 −0.883224
\(975\) 0 0
\(976\) −0.451876 −0.0144642
\(977\) 24.7525 0.791902 0.395951 0.918272i \(-0.370415\pi\)
0.395951 + 0.918272i \(0.370415\pi\)
\(978\) 5.72410 0.183036
\(979\) 21.1641 0.676408
\(980\) −4.17951 −0.133510
\(981\) −45.8815 −1.46488
\(982\) 30.7423 0.981025
\(983\) 4.55736 0.145357 0.0726786 0.997355i \(-0.476845\pi\)
0.0726786 + 0.997355i \(0.476845\pi\)
\(984\) 27.1728 0.866237
\(985\) −34.8152 −1.10930
\(986\) −13.3364 −0.424718
\(987\) −1.21328 −0.0386192
\(988\) 0 0
\(989\) −19.0820 −0.606773
\(990\) −63.0969 −2.00535
\(991\) −27.1963 −0.863919 −0.431960 0.901893i \(-0.642178\pi\)
−0.431960 + 0.901893i \(0.642178\pi\)
\(992\) −6.67085 −0.211800
\(993\) 41.5824 1.31958
\(994\) 3.71193 0.117735
\(995\) −67.0734 −2.12637
\(996\) 21.7936 0.690556
\(997\) −30.2274 −0.957312 −0.478656 0.878002i \(-0.658876\pi\)
−0.478656 + 0.878002i \(0.658876\pi\)
\(998\) −28.1102 −0.889812
\(999\) −18.6390 −0.589713
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 1183.2.a.p.1.2 6
7.6 odd 2 8281.2.a.ch.1.2 6
13.5 odd 4 1183.2.c.i.337.8 12
13.6 odd 12 91.2.q.a.36.4 12
13.8 odd 4 1183.2.c.i.337.5 12
13.11 odd 12 91.2.q.a.43.4 yes 12
13.12 even 2 1183.2.a.m.1.5 6
39.11 even 12 819.2.ct.a.316.3 12
39.32 even 12 819.2.ct.a.127.3 12
52.11 even 12 1456.2.cc.c.225.6 12
52.19 even 12 1456.2.cc.c.673.6 12
91.6 even 12 637.2.q.h.491.4 12
91.11 odd 12 637.2.u.h.30.3 12
91.19 even 12 637.2.u.i.361.3 12
91.24 even 12 637.2.u.i.30.3 12
91.32 odd 12 637.2.k.h.569.4 12
91.37 odd 12 637.2.k.h.459.3 12
91.45 even 12 637.2.k.g.569.4 12
91.58 odd 12 637.2.u.h.361.3 12
91.76 even 12 637.2.q.h.589.4 12
91.89 even 12 637.2.k.g.459.3 12
91.90 odd 2 8281.2.a.by.1.5 6
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
91.2.q.a.36.4 12 13.6 odd 12
91.2.q.a.43.4 yes 12 13.11 odd 12
637.2.k.g.459.3 12 91.89 even 12
637.2.k.g.569.4 12 91.45 even 12
637.2.k.h.459.3 12 91.37 odd 12
637.2.k.h.569.4 12 91.32 odd 12
637.2.q.h.491.4 12 91.6 even 12
637.2.q.h.589.4 12 91.76 even 12
637.2.u.h.30.3 12 91.11 odd 12
637.2.u.h.361.3 12 91.58 odd 12
637.2.u.i.30.3 12 91.24 even 12
637.2.u.i.361.3 12 91.19 even 12
819.2.ct.a.127.3 12 39.32 even 12
819.2.ct.a.316.3 12 39.11 even 12
1183.2.a.m.1.5 6 13.12 even 2
1183.2.a.p.1.2 6 1.1 even 1 trivial
1183.2.c.i.337.5 12 13.8 odd 4
1183.2.c.i.337.8 12 13.5 odd 4
1456.2.cc.c.225.6 12 52.11 even 12
1456.2.cc.c.673.6 12 52.19 even 12
8281.2.a.by.1.5 6 91.90 odd 2
8281.2.a.ch.1.2 6 7.6 odd 2