Properties

Label 1815.2.a.o.1.1
Level $1815$
Weight $2$
Character 1815.1
Self dual yes
Analytic conductor $14.493$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 1815 = 3 \cdot 5 \cdot 11^{2} \)
Weight: \( k \) \(=\) \( 2 \)
Character orbit: \([\chi]\) \(=\) 1815.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(14.4928479669\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: 4.4.725.1
Defining polynomial: \(x^{4} - x^{3} - 3 x^{2} + x + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 165)
Fricke sign: \(1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.1
Root \(-0.477260\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-2.77222 q^{2} +1.00000 q^{3} +5.68522 q^{4} -1.00000 q^{5} -2.77222 q^{6} +2.27759 q^{7} -10.2163 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-2.77222 q^{2} +1.00000 q^{3} +5.68522 q^{4} -1.00000 q^{5} -2.77222 q^{6} +2.27759 q^{7} -10.2163 q^{8} +1.00000 q^{9} +2.77222 q^{10} +5.68522 q^{12} +0.435737 q^{13} -6.31399 q^{14} -1.00000 q^{15} +16.9513 q^{16} -5.00000 q^{17} -2.77222 q^{18} -4.69596 q^{19} -5.68522 q^{20} +2.27759 q^{21} -0.845811 q^{23} -10.2163 q^{24} +1.00000 q^{25} -1.20796 q^{26} +1.00000 q^{27} +12.9486 q^{28} +2.65711 q^{29} +2.77222 q^{30} -4.66785 q^{31} -26.5602 q^{32} +13.8611 q^{34} -2.27759 q^{35} +5.68522 q^{36} -8.86239 q^{37} +13.0182 q^{38} +0.435737 q^{39} +10.2163 q^{40} +4.29417 q^{41} -6.31399 q^{42} -7.00317 q^{43} -1.00000 q^{45} +2.34478 q^{46} +0.468179 q^{47} +16.9513 q^{48} -1.81258 q^{49} -2.77222 q^{50} -5.00000 q^{51} +2.47726 q^{52} -10.8386 q^{53} -2.77222 q^{54} -23.2684 q^{56} -4.69596 q^{57} -7.36611 q^{58} +3.93598 q^{59} -5.68522 q^{60} -2.96549 q^{61} +12.9403 q^{62} +2.27759 q^{63} +39.7283 q^{64} -0.435737 q^{65} -2.47048 q^{67} -28.4261 q^{68} -0.845811 q^{69} +6.31399 q^{70} -11.3140 q^{71} -10.2163 q^{72} -8.42910 q^{73} +24.5685 q^{74} +1.00000 q^{75} -26.6975 q^{76} -1.20796 q^{78} +10.8707 q^{79} -16.9513 q^{80} +1.00000 q^{81} -11.9044 q^{82} +5.92050 q^{83} +12.9486 q^{84} +5.00000 q^{85} +19.4143 q^{86} +2.65711 q^{87} +5.89958 q^{89} +2.77222 q^{90} +0.992430 q^{91} -4.80862 q^{92} -4.66785 q^{93} -1.29790 q^{94} +4.69596 q^{95} -26.5602 q^{96} -8.64803 q^{97} +5.02487 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 5q^{2} + 4q^{3} + 9q^{4} - 4q^{5} - 5q^{6} + 2q^{7} - 15q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 5q^{2} + 4q^{3} + 9q^{4} - 4q^{5} - 5q^{6} + 2q^{7} - 15q^{8} + 4q^{9} + 5q^{10} + 9q^{12} - 3q^{13} - 5q^{14} - 4q^{15} + 15q^{16} - 20q^{17} - 5q^{18} - 3q^{19} - 9q^{20} + 2q^{21} - 5q^{23} - 15q^{24} + 4q^{25} + 6q^{26} + 4q^{27} - 3q^{28} - 5q^{29} + 5q^{30} - q^{31} - 30q^{32} + 25q^{34} - 2q^{35} + 9q^{36} - 7q^{37} + q^{38} - 3q^{39} + 15q^{40} - 20q^{41} - 5q^{42} + 2q^{43} - 4q^{45} - 7q^{46} - 20q^{47} + 15q^{48} + 8q^{49} - 5q^{50} - 20q^{51} + 7q^{52} + 6q^{53} - 5q^{54} + 10q^{56} - 3q^{57} - 21q^{58} - 5q^{59} - 9q^{60} + 7q^{61} + 12q^{62} + 2q^{63} + 49q^{64} + 3q^{65} - 13q^{67} - 45q^{68} - 5q^{69} + 5q^{70} - 25q^{71} - 15q^{72} - 23q^{73} + 7q^{74} + 4q^{75} + 7q^{76} + 6q^{78} - 15q^{80} + 4q^{81} + 11q^{82} - 33q^{83} - 3q^{84} + 20q^{85} - 12q^{86} - 5q^{87} + 16q^{89} + 5q^{90} - 24q^{91} - q^{93} + 17q^{94} + 3q^{95} - 30q^{96} - 25q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −2.77222 −1.96026 −0.980129 0.198362i \(-0.936438\pi\)
−0.980129 + 0.198362i \(0.936438\pi\)
\(3\) 1.00000 0.577350
\(4\) 5.68522 2.84261
\(5\) −1.00000 −0.447214
\(6\) −2.77222 −1.13176
\(7\) 2.27759 0.860849 0.430424 0.902627i \(-0.358364\pi\)
0.430424 + 0.902627i \(0.358364\pi\)
\(8\) −10.2163 −3.61199
\(9\) 1.00000 0.333333
\(10\) 2.77222 0.876654
\(11\) 0 0
\(12\) 5.68522 1.64118
\(13\) 0.435737 0.120852 0.0604258 0.998173i \(-0.480754\pi\)
0.0604258 + 0.998173i \(0.480754\pi\)
\(14\) −6.31399 −1.68748
\(15\) −1.00000 −0.258199
\(16\) 16.9513 4.23782
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) −2.77222 −0.653419
\(19\) −4.69596 −1.07733 −0.538663 0.842521i \(-0.681070\pi\)
−0.538663 + 0.842521i \(0.681070\pi\)
\(20\) −5.68522 −1.27125
\(21\) 2.27759 0.497011
\(22\) 0 0
\(23\) −0.845811 −0.176364 −0.0881819 0.996104i \(-0.528106\pi\)
−0.0881819 + 0.996104i \(0.528106\pi\)
\(24\) −10.2163 −2.08538
\(25\) 1.00000 0.200000
\(26\) −1.20796 −0.236900
\(27\) 1.00000 0.192450
\(28\) 12.9486 2.44706
\(29\) 2.65711 0.493413 0.246707 0.969090i \(-0.420652\pi\)
0.246707 + 0.969090i \(0.420652\pi\)
\(30\) 2.77222 0.506136
\(31\) −4.66785 −0.838370 −0.419185 0.907901i \(-0.637684\pi\)
−0.419185 + 0.907901i \(0.637684\pi\)
\(32\) −26.5602 −4.69523
\(33\) 0 0
\(34\) 13.8611 2.37716
\(35\) −2.27759 −0.384983
\(36\) 5.68522 0.947537
\(37\) −8.86239 −1.45697 −0.728484 0.685063i \(-0.759775\pi\)
−0.728484 + 0.685063i \(0.759775\pi\)
\(38\) 13.0182 2.11184
\(39\) 0.435737 0.0697737
\(40\) 10.2163 1.61533
\(41\) 4.29417 0.670637 0.335319 0.942105i \(-0.391156\pi\)
0.335319 + 0.942105i \(0.391156\pi\)
\(42\) −6.31399 −0.974270
\(43\) −7.00317 −1.06797 −0.533986 0.845493i \(-0.679307\pi\)
−0.533986 + 0.845493i \(0.679307\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 2.34478 0.345718
\(47\) 0.468179 0.0682909 0.0341455 0.999417i \(-0.489129\pi\)
0.0341455 + 0.999417i \(0.489129\pi\)
\(48\) 16.9513 2.44671
\(49\) −1.81258 −0.258940
\(50\) −2.77222 −0.392052
\(51\) −5.00000 −0.700140
\(52\) 2.47726 0.343534
\(53\) −10.8386 −1.48880 −0.744399 0.667735i \(-0.767264\pi\)
−0.744399 + 0.667735i \(0.767264\pi\)
\(54\) −2.77222 −0.377252
\(55\) 0 0
\(56\) −23.2684 −3.10938
\(57\) −4.69596 −0.621995
\(58\) −7.36611 −0.967217
\(59\) 3.93598 0.512421 0.256211 0.966621i \(-0.417526\pi\)
0.256211 + 0.966621i \(0.417526\pi\)
\(60\) −5.68522 −0.733959
\(61\) −2.96549 −0.379692 −0.189846 0.981814i \(-0.560799\pi\)
−0.189846 + 0.981814i \(0.560799\pi\)
\(62\) 12.9403 1.64342
\(63\) 2.27759 0.286950
\(64\) 39.7283 4.96604
\(65\) −0.435737 −0.0540465
\(66\) 0 0
\(67\) −2.47048 −0.301817 −0.150909 0.988548i \(-0.548220\pi\)
−0.150909 + 0.988548i \(0.548220\pi\)
\(68\) −28.4261 −3.44717
\(69\) −0.845811 −0.101824
\(70\) 6.31399 0.754666
\(71\) −11.3140 −1.34272 −0.671362 0.741130i \(-0.734290\pi\)
−0.671362 + 0.741130i \(0.734290\pi\)
\(72\) −10.2163 −1.20400
\(73\) −8.42910 −0.986552 −0.493276 0.869873i \(-0.664201\pi\)
−0.493276 + 0.869873i \(0.664201\pi\)
\(74\) 24.5685 2.85603
\(75\) 1.00000 0.115470
\(76\) −26.6975 −3.06242
\(77\) 0 0
\(78\) −1.20796 −0.136775
\(79\) 10.8707 1.22305 0.611524 0.791226i \(-0.290557\pi\)
0.611524 + 0.791226i \(0.290557\pi\)
\(80\) −16.9513 −1.89521
\(81\) 1.00000 0.111111
\(82\) −11.9044 −1.31462
\(83\) 5.92050 0.649859 0.324930 0.945738i \(-0.394659\pi\)
0.324930 + 0.945738i \(0.394659\pi\)
\(84\) 12.9486 1.41281
\(85\) 5.00000 0.542326
\(86\) 19.4143 2.09350
\(87\) 2.65711 0.284872
\(88\) 0 0
\(89\) 5.89958 0.625354 0.312677 0.949859i \(-0.398774\pi\)
0.312677 + 0.949859i \(0.398774\pi\)
\(90\) 2.77222 0.292218
\(91\) 0.992430 0.104035
\(92\) −4.80862 −0.501333
\(93\) −4.66785 −0.484033
\(94\) −1.29790 −0.133868
\(95\) 4.69596 0.481795
\(96\) −26.5602 −2.71079
\(97\) −8.64803 −0.878074 −0.439037 0.898469i \(-0.644680\pi\)
−0.439037 + 0.898469i \(0.644680\pi\)
\(98\) 5.02487 0.507589
\(99\) 0 0
\(100\) 5.68522 0.568522
\(101\) 5.68126 0.565307 0.282653 0.959222i \(-0.408785\pi\)
0.282653 + 0.959222i \(0.408785\pi\)
\(102\) 13.8611 1.37245
\(103\) −1.29613 −0.127711 −0.0638557 0.997959i \(-0.520340\pi\)
−0.0638557 + 0.997959i \(0.520340\pi\)
\(104\) −4.45160 −0.436515
\(105\) −2.27759 −0.222270
\(106\) 30.0471 2.91843
\(107\) 12.2008 1.17949 0.589746 0.807589i \(-0.299228\pi\)
0.589746 + 0.807589i \(0.299228\pi\)
\(108\) 5.68522 0.547061
\(109\) 12.1644 1.16514 0.582568 0.812782i \(-0.302048\pi\)
0.582568 + 0.812782i \(0.302048\pi\)
\(110\) 0 0
\(111\) −8.86239 −0.841181
\(112\) 38.6081 3.64812
\(113\) 0.142052 0.0133632 0.00668158 0.999978i \(-0.497873\pi\)
0.00668158 + 0.999978i \(0.497873\pi\)
\(114\) 13.0182 1.21927
\(115\) 0.845811 0.0788723
\(116\) 15.1063 1.40258
\(117\) 0.435737 0.0402839
\(118\) −10.9114 −1.00448
\(119\) −11.3880 −1.04393
\(120\) 10.2163 0.932612
\(121\) 0 0
\(122\) 8.22100 0.744294
\(123\) 4.29417 0.387193
\(124\) −26.5377 −2.38316
\(125\) −1.00000 −0.0894427
\(126\) −6.31399 −0.562495
\(127\) 0.503713 0.0446973 0.0223486 0.999750i \(-0.492886\pi\)
0.0223486 + 0.999750i \(0.492886\pi\)
\(128\) −57.0153 −5.03949
\(129\) −7.00317 −0.616594
\(130\) 1.20796 0.105945
\(131\) −19.1098 −1.66963 −0.834816 0.550529i \(-0.814426\pi\)
−0.834816 + 0.550529i \(0.814426\pi\)
\(132\) 0 0
\(133\) −10.6955 −0.927415
\(134\) 6.84872 0.591640
\(135\) −1.00000 −0.0860663
\(136\) 51.0813 4.38018
\(137\) 12.3448 1.05469 0.527343 0.849653i \(-0.323188\pi\)
0.527343 + 0.849653i \(0.323188\pi\)
\(138\) 2.34478 0.199601
\(139\) 7.73236 0.655850 0.327925 0.944704i \(-0.393651\pi\)
0.327925 + 0.944704i \(0.393651\pi\)
\(140\) −12.9486 −1.09436
\(141\) 0.468179 0.0394278
\(142\) 31.3649 2.63208
\(143\) 0 0
\(144\) 16.9513 1.41261
\(145\) −2.65711 −0.220661
\(146\) 23.3673 1.93390
\(147\) −1.81258 −0.149499
\(148\) −50.3847 −4.14159
\(149\) −7.60895 −0.623350 −0.311675 0.950189i \(-0.600890\pi\)
−0.311675 + 0.950189i \(0.600890\pi\)
\(150\) −2.77222 −0.226351
\(151\) 2.67425 0.217627 0.108814 0.994062i \(-0.465295\pi\)
0.108814 + 0.994062i \(0.465295\pi\)
\(152\) 47.9751 3.89129
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 4.66785 0.374931
\(156\) 2.47726 0.198340
\(157\) 20.2000 1.61213 0.806067 0.591825i \(-0.201592\pi\)
0.806067 + 0.591825i \(0.201592\pi\)
\(158\) −30.1360 −2.39749
\(159\) −10.8386 −0.859558
\(160\) 26.5602 2.09977
\(161\) −1.92641 −0.151823
\(162\) −2.77222 −0.217806
\(163\) −9.79582 −0.767268 −0.383634 0.923485i \(-0.625328\pi\)
−0.383634 + 0.923485i \(0.625328\pi\)
\(164\) 24.4133 1.90636
\(165\) 0 0
\(166\) −16.4129 −1.27389
\(167\) −25.5776 −1.97925 −0.989627 0.143658i \(-0.954114\pi\)
−0.989627 + 0.143658i \(0.954114\pi\)
\(168\) −23.2684 −1.79520
\(169\) −12.8101 −0.985395
\(170\) −13.8611 −1.06310
\(171\) −4.69596 −0.359109
\(172\) −39.8145 −3.03583
\(173\) 6.75406 0.513502 0.256751 0.966478i \(-0.417348\pi\)
0.256751 + 0.966478i \(0.417348\pi\)
\(174\) −7.36611 −0.558423
\(175\) 2.27759 0.172170
\(176\) 0 0
\(177\) 3.93598 0.295846
\(178\) −16.3550 −1.22586
\(179\) 6.52195 0.487473 0.243737 0.969841i \(-0.421627\pi\)
0.243737 + 0.969841i \(0.421627\pi\)
\(180\) −5.68522 −0.423751
\(181\) 13.7522 1.02219 0.511095 0.859524i \(-0.329240\pi\)
0.511095 + 0.859524i \(0.329240\pi\)
\(182\) −2.75124 −0.203935
\(183\) −2.96549 −0.219215
\(184\) 8.64102 0.637024
\(185\) 8.86239 0.651576
\(186\) 12.9403 0.948830
\(187\) 0 0
\(188\) 2.66170 0.194124
\(189\) 2.27759 0.165670
\(190\) −13.0182 −0.944442
\(191\) −19.4360 −1.40634 −0.703171 0.711020i \(-0.748233\pi\)
−0.703171 + 0.711020i \(0.748233\pi\)
\(192\) 39.7283 2.86715
\(193\) −22.6124 −1.62767 −0.813836 0.581094i \(-0.802625\pi\)
−0.813836 + 0.581094i \(0.802625\pi\)
\(194\) 23.9743 1.72125
\(195\) −0.435737 −0.0312038
\(196\) −10.3049 −0.736065
\(197\) −23.0300 −1.64082 −0.820410 0.571776i \(-0.806255\pi\)
−0.820410 + 0.571776i \(0.806255\pi\)
\(198\) 0 0
\(199\) −19.6216 −1.39094 −0.695468 0.718557i \(-0.744803\pi\)
−0.695468 + 0.718557i \(0.744803\pi\)
\(200\) −10.2163 −0.722398
\(201\) −2.47048 −0.174254
\(202\) −15.7497 −1.10815
\(203\) 6.05181 0.424754
\(204\) −28.4261 −1.99023
\(205\) −4.29417 −0.299918
\(206\) 3.59316 0.250347
\(207\) −0.845811 −0.0587879
\(208\) 7.38630 0.512148
\(209\) 0 0
\(210\) 6.31399 0.435707
\(211\) −0.0499252 −0.00343699 −0.00171850 0.999999i \(-0.500547\pi\)
−0.00171850 + 0.999999i \(0.500547\pi\)
\(212\) −61.6199 −4.23207
\(213\) −11.3140 −0.775222
\(214\) −33.8232 −2.31211
\(215\) 7.00317 0.477612
\(216\) −10.2163 −0.695128
\(217\) −10.6314 −0.721710
\(218\) −33.7223 −2.28397
\(219\) −8.42910 −0.569586
\(220\) 0 0
\(221\) −2.17868 −0.146554
\(222\) 24.5685 1.64893
\(223\) 0.754962 0.0505560 0.0252780 0.999680i \(-0.491953\pi\)
0.0252780 + 0.999680i \(0.491953\pi\)
\(224\) −60.4934 −4.04188
\(225\) 1.00000 0.0666667
\(226\) −0.393801 −0.0261952
\(227\) −8.28399 −0.549828 −0.274914 0.961469i \(-0.588649\pi\)
−0.274914 + 0.961469i \(0.588649\pi\)
\(228\) −26.6975 −1.76809
\(229\) −2.87622 −0.190066 −0.0950330 0.995474i \(-0.530296\pi\)
−0.0950330 + 0.995474i \(0.530296\pi\)
\(230\) −2.34478 −0.154610
\(231\) 0 0
\(232\) −27.1457 −1.78220
\(233\) 5.17868 0.339267 0.169633 0.985507i \(-0.445742\pi\)
0.169633 + 0.985507i \(0.445742\pi\)
\(234\) −1.20796 −0.0789668
\(235\) −0.468179 −0.0305406
\(236\) 22.3769 1.45661
\(237\) 10.8707 0.706127
\(238\) 31.5700 2.04638
\(239\) −10.8328 −0.700714 −0.350357 0.936616i \(-0.613940\pi\)
−0.350357 + 0.936616i \(0.613940\pi\)
\(240\) −16.9513 −1.09420
\(241\) 2.96526 0.191009 0.0955045 0.995429i \(-0.469554\pi\)
0.0955045 + 0.995429i \(0.469554\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −16.8595 −1.07932
\(245\) 1.81258 0.115801
\(246\) −11.9044 −0.758997
\(247\) −2.04620 −0.130197
\(248\) 47.6879 3.02818
\(249\) 5.92050 0.375196
\(250\) 2.77222 0.175331
\(251\) −17.6144 −1.11181 −0.555906 0.831245i \(-0.687629\pi\)
−0.555906 + 0.831245i \(0.687629\pi\)
\(252\) 12.9486 0.815685
\(253\) 0 0
\(254\) −1.39640 −0.0876182
\(255\) 5.00000 0.313112
\(256\) 78.6025 4.91266
\(257\) 20.9647 1.30774 0.653871 0.756606i \(-0.273144\pi\)
0.653871 + 0.756606i \(0.273144\pi\)
\(258\) 19.4143 1.20868
\(259\) −20.1849 −1.25423
\(260\) −2.47726 −0.153633
\(261\) 2.65711 0.164471
\(262\) 52.9766 3.27291
\(263\) −8.43471 −0.520107 −0.260053 0.965594i \(-0.583740\pi\)
−0.260053 + 0.965594i \(0.583740\pi\)
\(264\) 0 0
\(265\) 10.8386 0.665811
\(266\) 29.6502 1.81797
\(267\) 5.89958 0.361049
\(268\) −14.0452 −0.857949
\(269\) 9.64346 0.587972 0.293986 0.955810i \(-0.405018\pi\)
0.293986 + 0.955810i \(0.405018\pi\)
\(270\) 2.77222 0.168712
\(271\) −10.2278 −0.621293 −0.310647 0.950525i \(-0.600546\pi\)
−0.310647 + 0.950525i \(0.600546\pi\)
\(272\) −84.7564 −5.13911
\(273\) 0.992430 0.0600646
\(274\) −34.2225 −2.06746
\(275\) 0 0
\(276\) −4.80862 −0.289445
\(277\) 5.83903 0.350833 0.175417 0.984494i \(-0.443873\pi\)
0.175417 + 0.984494i \(0.443873\pi\)
\(278\) −21.4358 −1.28563
\(279\) −4.66785 −0.279457
\(280\) 23.2684 1.39056
\(281\) −14.4139 −0.859858 −0.429929 0.902863i \(-0.641461\pi\)
−0.429929 + 0.902863i \(0.641461\pi\)
\(282\) −1.29790 −0.0772886
\(283\) 10.6334 0.632093 0.316046 0.948744i \(-0.397644\pi\)
0.316046 + 0.948744i \(0.397644\pi\)
\(284\) −64.3225 −3.81684
\(285\) 4.69596 0.278164
\(286\) 0 0
\(287\) 9.78037 0.577317
\(288\) −26.5602 −1.56508
\(289\) 8.00000 0.470588
\(290\) 7.36611 0.432553
\(291\) −8.64803 −0.506957
\(292\) −47.9213 −2.80438
\(293\) 8.39576 0.490485 0.245243 0.969462i \(-0.421132\pi\)
0.245243 + 0.969462i \(0.421132\pi\)
\(294\) 5.02487 0.293057
\(295\) −3.93598 −0.229162
\(296\) 90.5404 5.26256
\(297\) 0 0
\(298\) 21.0937 1.22193
\(299\) −0.368551 −0.0213139
\(300\) 5.68522 0.328236
\(301\) −15.9504 −0.919363
\(302\) −7.41362 −0.426606
\(303\) 5.68126 0.326380
\(304\) −79.6025 −4.56552
\(305\) 2.96549 0.169803
\(306\) 13.8611 0.792387
\(307\) −4.51902 −0.257914 −0.128957 0.991650i \(-0.541163\pi\)
−0.128957 + 0.991650i \(0.541163\pi\)
\(308\) 0 0
\(309\) −1.29613 −0.0737343
\(310\) −12.9403 −0.734961
\(311\) 23.9970 1.36074 0.680372 0.732867i \(-0.261818\pi\)
0.680372 + 0.732867i \(0.261818\pi\)
\(312\) −4.45160 −0.252022
\(313\) −25.8686 −1.46218 −0.731090 0.682281i \(-0.760988\pi\)
−0.731090 + 0.682281i \(0.760988\pi\)
\(314\) −55.9988 −3.16020
\(315\) −2.27759 −0.128328
\(316\) 61.8022 3.47665
\(317\) 2.90319 0.163060 0.0815298 0.996671i \(-0.474019\pi\)
0.0815298 + 0.996671i \(0.474019\pi\)
\(318\) 30.0471 1.68496
\(319\) 0 0
\(320\) −39.7283 −2.22088
\(321\) 12.2008 0.680980
\(322\) 5.34044 0.297611
\(323\) 23.4798 1.30645
\(324\) 5.68522 0.315846
\(325\) 0.435737 0.0241703
\(326\) 27.1562 1.50404
\(327\) 12.1644 0.672691
\(328\) −43.8703 −2.42233
\(329\) 1.06632 0.0587881
\(330\) 0 0
\(331\) 10.9837 0.603720 0.301860 0.953352i \(-0.402392\pi\)
0.301860 + 0.953352i \(0.402392\pi\)
\(332\) 33.6593 1.84730
\(333\) −8.86239 −0.485656
\(334\) 70.9068 3.87985
\(335\) 2.47048 0.134977
\(336\) 38.6081 2.10624
\(337\) −12.8462 −0.699775 −0.349887 0.936792i \(-0.613780\pi\)
−0.349887 + 0.936792i \(0.613780\pi\)
\(338\) 35.5125 1.93163
\(339\) 0.142052 0.00771522
\(340\) 28.4261 1.54162
\(341\) 0 0
\(342\) 13.0182 0.703946
\(343\) −20.0715 −1.08376
\(344\) 71.5461 3.85751
\(345\) 0.845811 0.0455369
\(346\) −18.7238 −1.00660
\(347\) 1.88971 0.101445 0.0507225 0.998713i \(-0.483848\pi\)
0.0507225 + 0.998713i \(0.483848\pi\)
\(348\) 15.1063 0.809781
\(349\) −23.7787 −1.27284 −0.636422 0.771341i \(-0.719586\pi\)
−0.636422 + 0.771341i \(0.719586\pi\)
\(350\) −6.31399 −0.337497
\(351\) 0.435737 0.0232579
\(352\) 0 0
\(353\) −20.2294 −1.07670 −0.538352 0.842720i \(-0.680953\pi\)
−0.538352 + 0.842720i \(0.680953\pi\)
\(354\) −10.9114 −0.579935
\(355\) 11.3140 0.600484
\(356\) 33.5404 1.77764
\(357\) −11.3880 −0.602715
\(358\) −18.0803 −0.955573
\(359\) 18.5897 0.981129 0.490565 0.871405i \(-0.336791\pi\)
0.490565 + 0.871405i \(0.336791\pi\)
\(360\) 10.2163 0.538444
\(361\) 3.05200 0.160632
\(362\) −38.1241 −2.00376
\(363\) 0 0
\(364\) 5.64219 0.295731
\(365\) 8.42910 0.441199
\(366\) 8.22100 0.429718
\(367\) 5.90811 0.308401 0.154200 0.988040i \(-0.450720\pi\)
0.154200 + 0.988040i \(0.450720\pi\)
\(368\) −14.3376 −0.747398
\(369\) 4.29417 0.223546
\(370\) −24.5685 −1.27726
\(371\) −24.6859 −1.28163
\(372\) −26.5377 −1.37592
\(373\) 1.66992 0.0864650 0.0432325 0.999065i \(-0.486234\pi\)
0.0432325 + 0.999065i \(0.486234\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −4.78303 −0.246666
\(377\) 1.15780 0.0596298
\(378\) −6.31399 −0.324757
\(379\) 8.43518 0.433286 0.216643 0.976251i \(-0.430489\pi\)
0.216643 + 0.976251i \(0.430489\pi\)
\(380\) 26.6975 1.36956
\(381\) 0.503713 0.0258060
\(382\) 53.8810 2.75679
\(383\) 21.8815 1.11809 0.559047 0.829136i \(-0.311167\pi\)
0.559047 + 0.829136i \(0.311167\pi\)
\(384\) −57.0153 −2.90955
\(385\) 0 0
\(386\) 62.6865 3.19066
\(387\) −7.00317 −0.355991
\(388\) −49.1660 −2.49602
\(389\) −0.392092 −0.0198799 −0.00993993 0.999951i \(-0.503164\pi\)
−0.00993993 + 0.999951i \(0.503164\pi\)
\(390\) 1.20796 0.0611674
\(391\) 4.22906 0.213873
\(392\) 18.5178 0.935288
\(393\) −19.1098 −0.963962
\(394\) 63.8443 3.21643
\(395\) −10.8707 −0.546963
\(396\) 0 0
\(397\) 35.7823 1.79586 0.897932 0.440134i \(-0.145069\pi\)
0.897932 + 0.440134i \(0.145069\pi\)
\(398\) 54.3954 2.72659
\(399\) −10.6955 −0.535443
\(400\) 16.9513 0.847564
\(401\) 30.4165 1.51893 0.759463 0.650551i \(-0.225462\pi\)
0.759463 + 0.650551i \(0.225462\pi\)
\(402\) 6.84872 0.341583
\(403\) −2.03395 −0.101318
\(404\) 32.2992 1.60695
\(405\) −1.00000 −0.0496904
\(406\) −16.7770 −0.832627
\(407\) 0 0
\(408\) 51.0813 2.52890
\(409\) 16.2697 0.804486 0.402243 0.915533i \(-0.368231\pi\)
0.402243 + 0.915533i \(0.368231\pi\)
\(410\) 11.9044 0.587917
\(411\) 12.3448 0.608923
\(412\) −7.36878 −0.363034
\(413\) 8.96456 0.441117
\(414\) 2.34478 0.115239
\(415\) −5.92050 −0.290626
\(416\) −11.5733 −0.567427
\(417\) 7.73236 0.378655
\(418\) 0 0
\(419\) −40.0703 −1.95756 −0.978781 0.204910i \(-0.934310\pi\)
−0.978781 + 0.204910i \(0.934310\pi\)
\(420\) −12.9486 −0.631827
\(421\) −19.3943 −0.945219 −0.472609 0.881272i \(-0.656688\pi\)
−0.472609 + 0.881272i \(0.656688\pi\)
\(422\) 0.138404 0.00673739
\(423\) 0.468179 0.0227636
\(424\) 110.730 5.37753
\(425\) −5.00000 −0.242536
\(426\) 31.3649 1.51963
\(427\) −6.75417 −0.326857
\(428\) 69.3640 3.35284
\(429\) 0 0
\(430\) −19.4143 −0.936243
\(431\) −33.9444 −1.63505 −0.817523 0.575896i \(-0.804653\pi\)
−0.817523 + 0.575896i \(0.804653\pi\)
\(432\) 16.9513 0.815569
\(433\) −13.1155 −0.630290 −0.315145 0.949044i \(-0.602053\pi\)
−0.315145 + 0.949044i \(0.602053\pi\)
\(434\) 29.4727 1.41474
\(435\) −2.65711 −0.127399
\(436\) 69.1571 3.31202
\(437\) 3.97189 0.190001
\(438\) 23.3673 1.11654
\(439\) 9.64731 0.460441 0.230220 0.973138i \(-0.426055\pi\)
0.230220 + 0.973138i \(0.426055\pi\)
\(440\) 0 0
\(441\) −1.81258 −0.0863133
\(442\) 6.03980 0.287284
\(443\) −8.91410 −0.423521 −0.211761 0.977322i \(-0.567920\pi\)
−0.211761 + 0.977322i \(0.567920\pi\)
\(444\) −50.3847 −2.39115
\(445\) −5.89958 −0.279667
\(446\) −2.09292 −0.0991028
\(447\) −7.60895 −0.359891
\(448\) 90.4849 4.27501
\(449\) −12.4368 −0.586931 −0.293465 0.955970i \(-0.594809\pi\)
−0.293465 + 0.955970i \(0.594809\pi\)
\(450\) −2.77222 −0.130684
\(451\) 0 0
\(452\) 0.807599 0.0379863
\(453\) 2.67425 0.125647
\(454\) 22.9651 1.07780
\(455\) −0.992430 −0.0465259
\(456\) 47.9751 2.24664
\(457\) 38.1583 1.78497 0.892484 0.451078i \(-0.148960\pi\)
0.892484 + 0.451078i \(0.148960\pi\)
\(458\) 7.97353 0.372578
\(459\) −5.00000 −0.233380
\(460\) 4.80862 0.224203
\(461\) −34.3847 −1.60145 −0.800726 0.599030i \(-0.795553\pi\)
−0.800726 + 0.599030i \(0.795553\pi\)
\(462\) 0 0
\(463\) −40.2561 −1.87086 −0.935430 0.353511i \(-0.884988\pi\)
−0.935430 + 0.353511i \(0.884988\pi\)
\(464\) 45.0415 2.09100
\(465\) 4.66785 0.216466
\(466\) −14.3565 −0.665051
\(467\) 15.1161 0.699490 0.349745 0.936845i \(-0.386268\pi\)
0.349745 + 0.936845i \(0.386268\pi\)
\(468\) 2.47726 0.114511
\(469\) −5.62674 −0.259819
\(470\) 1.29790 0.0598675
\(471\) 20.2000 0.930766
\(472\) −40.2110 −1.85086
\(473\) 0 0
\(474\) −30.1360 −1.38419
\(475\) −4.69596 −0.215465
\(476\) −64.7430 −2.96749
\(477\) −10.8386 −0.496266
\(478\) 30.0309 1.37358
\(479\) 28.8045 1.31611 0.658055 0.752970i \(-0.271379\pi\)
0.658055 + 0.752970i \(0.271379\pi\)
\(480\) 26.5602 1.21230
\(481\) −3.86167 −0.176077
\(482\) −8.22035 −0.374427
\(483\) −1.92641 −0.0876548
\(484\) 0 0
\(485\) 8.64803 0.392687
\(486\) −2.77222 −0.125751
\(487\) 28.8525 1.30743 0.653716 0.756740i \(-0.273209\pi\)
0.653716 + 0.756740i \(0.273209\pi\)
\(488\) 30.2962 1.37144
\(489\) −9.79582 −0.442982
\(490\) −5.02487 −0.227001
\(491\) 14.9367 0.674084 0.337042 0.941490i \(-0.390574\pi\)
0.337042 + 0.941490i \(0.390574\pi\)
\(492\) 24.4133 1.10064
\(493\) −13.2856 −0.598351
\(494\) 5.67253 0.255219
\(495\) 0 0
\(496\) −79.1260 −3.55286
\(497\) −25.7686 −1.15588
\(498\) −16.4129 −0.735481
\(499\) 22.1990 0.993764 0.496882 0.867818i \(-0.334478\pi\)
0.496882 + 0.867818i \(0.334478\pi\)
\(500\) −5.68522 −0.254251
\(501\) −25.5776 −1.14272
\(502\) 48.8311 2.17944
\(503\) −0.155356 −0.00692698 −0.00346349 0.999994i \(-0.501102\pi\)
−0.00346349 + 0.999994i \(0.501102\pi\)
\(504\) −23.2684 −1.03646
\(505\) −5.68126 −0.252813
\(506\) 0 0
\(507\) −12.8101 −0.568918
\(508\) 2.86372 0.127057
\(509\) −18.6375 −0.826095 −0.413047 0.910710i \(-0.635536\pi\)
−0.413047 + 0.910710i \(0.635536\pi\)
\(510\) −13.8611 −0.613780
\(511\) −19.1980 −0.849272
\(512\) −103.873 −4.59058
\(513\) −4.69596 −0.207332
\(514\) −58.1188 −2.56351
\(515\) 1.29613 0.0571143
\(516\) −39.8145 −1.75274
\(517\) 0 0
\(518\) 55.9571 2.45861
\(519\) 6.75406 0.296470
\(520\) 4.45160 0.195215
\(521\) 31.3888 1.37517 0.687585 0.726104i \(-0.258671\pi\)
0.687585 + 0.726104i \(0.258671\pi\)
\(522\) −7.36611 −0.322406
\(523\) 23.9841 1.04875 0.524377 0.851486i \(-0.324298\pi\)
0.524377 + 0.851486i \(0.324298\pi\)
\(524\) −108.643 −4.74611
\(525\) 2.27759 0.0994022
\(526\) 23.3829 1.01954
\(527\) 23.3392 1.01667
\(528\) 0 0
\(529\) −22.2846 −0.968896
\(530\) −30.0471 −1.30516
\(531\) 3.93598 0.170807
\(532\) −60.8061 −2.63628
\(533\) 1.87113 0.0810476
\(534\) −16.3550 −0.707748
\(535\) −12.2008 −0.527485
\(536\) 25.2390 1.09016
\(537\) 6.52195 0.281443
\(538\) −26.7338 −1.15258
\(539\) 0 0
\(540\) −5.68522 −0.244653
\(541\) −9.97322 −0.428782 −0.214391 0.976748i \(-0.568777\pi\)
−0.214391 + 0.976748i \(0.568777\pi\)
\(542\) 28.3537 1.21789
\(543\) 13.7522 0.590162
\(544\) 132.801 5.69380
\(545\) −12.1644 −0.521064
\(546\) −2.75124 −0.117742
\(547\) 40.0167 1.71099 0.855496 0.517809i \(-0.173252\pi\)
0.855496 + 0.517809i \(0.173252\pi\)
\(548\) 70.1828 2.99806
\(549\) −2.96549 −0.126564
\(550\) 0 0
\(551\) −12.4777 −0.531567
\(552\) 8.64102 0.367786
\(553\) 24.7590 1.05286
\(554\) −16.1871 −0.687724
\(555\) 8.86239 0.376188
\(556\) 43.9601 1.86433
\(557\) 7.76046 0.328821 0.164411 0.986392i \(-0.447428\pi\)
0.164411 + 0.986392i \(0.447428\pi\)
\(558\) 12.9403 0.547807
\(559\) −3.05154 −0.129066
\(560\) −38.6081 −1.63149
\(561\) 0 0
\(562\) 39.9584 1.68554
\(563\) −18.4355 −0.776964 −0.388482 0.921456i \(-0.627000\pi\)
−0.388482 + 0.921456i \(0.627000\pi\)
\(564\) 2.66170 0.112078
\(565\) −0.142052 −0.00597619
\(566\) −29.4783 −1.23906
\(567\) 2.27759 0.0956498
\(568\) 115.587 4.84990
\(569\) −11.0239 −0.462148 −0.231074 0.972936i \(-0.574224\pi\)
−0.231074 + 0.972936i \(0.574224\pi\)
\(570\) −13.0182 −0.545274
\(571\) 38.1338 1.59585 0.797926 0.602756i \(-0.205931\pi\)
0.797926 + 0.602756i \(0.205931\pi\)
\(572\) 0 0
\(573\) −19.4360 −0.811952
\(574\) −27.1134 −1.13169
\(575\) −0.845811 −0.0352728
\(576\) 39.7283 1.65535
\(577\) 11.0122 0.458446 0.229223 0.973374i \(-0.426382\pi\)
0.229223 + 0.973374i \(0.426382\pi\)
\(578\) −22.1778 −0.922474
\(579\) −22.6124 −0.939737
\(580\) −15.1063 −0.627253
\(581\) 13.4845 0.559430
\(582\) 23.9743 0.993765
\(583\) 0 0
\(584\) 86.1138 3.56341
\(585\) −0.435737 −0.0180155
\(586\) −23.2749 −0.961478
\(587\) 11.8440 0.488853 0.244427 0.969668i \(-0.421400\pi\)
0.244427 + 0.969668i \(0.421400\pi\)
\(588\) −10.3049 −0.424967
\(589\) 21.9200 0.903198
\(590\) 10.9114 0.449216
\(591\) −23.0300 −0.947327
\(592\) −150.229 −6.17437
\(593\) −9.25062 −0.379877 −0.189939 0.981796i \(-0.560829\pi\)
−0.189939 + 0.981796i \(0.560829\pi\)
\(594\) 0 0
\(595\) 11.3880 0.466861
\(596\) −43.2586 −1.77194
\(597\) −19.6216 −0.803058
\(598\) 1.02171 0.0417807
\(599\) 23.6834 0.967678 0.483839 0.875157i \(-0.339242\pi\)
0.483839 + 0.875157i \(0.339242\pi\)
\(600\) −10.2163 −0.417077
\(601\) −36.2731 −1.47961 −0.739806 0.672821i \(-0.765083\pi\)
−0.739806 + 0.672821i \(0.765083\pi\)
\(602\) 44.2179 1.80219
\(603\) −2.47048 −0.100606
\(604\) 15.2037 0.618630
\(605\) 0 0
\(606\) −15.7497 −0.639789
\(607\) 24.5068 0.994699 0.497350 0.867550i \(-0.334307\pi\)
0.497350 + 0.867550i \(0.334307\pi\)
\(608\) 124.726 5.05830
\(609\) 6.05181 0.245232
\(610\) −8.22100 −0.332858
\(611\) 0.204003 0.00825307
\(612\) −28.4261 −1.14906
\(613\) −4.64498 −0.187609 −0.0938044 0.995591i \(-0.529903\pi\)
−0.0938044 + 0.995591i \(0.529903\pi\)
\(614\) 12.5277 0.505578
\(615\) −4.29417 −0.173158
\(616\) 0 0
\(617\) 41.7419 1.68047 0.840233 0.542226i \(-0.182418\pi\)
0.840233 + 0.542226i \(0.182418\pi\)
\(618\) 3.59316 0.144538
\(619\) −44.2019 −1.77663 −0.888313 0.459239i \(-0.848122\pi\)
−0.888313 + 0.459239i \(0.848122\pi\)
\(620\) 26.5377 1.06578
\(621\) −0.845811 −0.0339412
\(622\) −66.5250 −2.66741
\(623\) 13.4368 0.538335
\(624\) 7.38630 0.295689
\(625\) 1.00000 0.0400000
\(626\) 71.7136 2.86625
\(627\) 0 0
\(628\) 114.841 4.58267
\(629\) 44.3120 1.76683
\(630\) 6.31399 0.251555
\(631\) −16.1161 −0.641572 −0.320786 0.947152i \(-0.603947\pi\)
−0.320786 + 0.947152i \(0.603947\pi\)
\(632\) −111.058 −4.41764
\(633\) −0.0499252 −0.00198435
\(634\) −8.04830 −0.319639
\(635\) −0.503713 −0.0199892
\(636\) −61.6199 −2.44339
\(637\) −0.789807 −0.0312933
\(638\) 0 0
\(639\) −11.3140 −0.447575
\(640\) 57.0153 2.25373
\(641\) −9.20985 −0.363767 −0.181884 0.983320i \(-0.558219\pi\)
−0.181884 + 0.983320i \(0.558219\pi\)
\(642\) −33.8232 −1.33490
\(643\) 4.01530 0.158348 0.0791741 0.996861i \(-0.474772\pi\)
0.0791741 + 0.996861i \(0.474772\pi\)
\(644\) −10.9521 −0.431572
\(645\) 7.00317 0.275749
\(646\) −65.0912 −2.56098
\(647\) 14.2693 0.560984 0.280492 0.959856i \(-0.409502\pi\)
0.280492 + 0.959856i \(0.409502\pi\)
\(648\) −10.2163 −0.401332
\(649\) 0 0
\(650\) −1.20796 −0.0473801
\(651\) −10.6314 −0.416679
\(652\) −55.6914 −2.18104
\(653\) −6.52172 −0.255215 −0.127607 0.991825i \(-0.540730\pi\)
−0.127607 + 0.991825i \(0.540730\pi\)
\(654\) −33.7223 −1.31865
\(655\) 19.1098 0.746682
\(656\) 72.7917 2.84204
\(657\) −8.42910 −0.328851
\(658\) −2.95608 −0.115240
\(659\) −6.74928 −0.262915 −0.131457 0.991322i \(-0.541966\pi\)
−0.131457 + 0.991322i \(0.541966\pi\)
\(660\) 0 0
\(661\) 9.65248 0.375438 0.187719 0.982223i \(-0.439891\pi\)
0.187719 + 0.982223i \(0.439891\pi\)
\(662\) −30.4493 −1.18345
\(663\) −2.17868 −0.0846131
\(664\) −60.4853 −2.34728
\(665\) 10.6955 0.414752
\(666\) 24.5685 0.952011
\(667\) −2.24741 −0.0870202
\(668\) −145.414 −5.62625
\(669\) 0.754962 0.0291885
\(670\) −6.84872 −0.264589
\(671\) 0 0
\(672\) −60.4934 −2.33358
\(673\) 28.3594 1.09318 0.546588 0.837402i \(-0.315926\pi\)
0.546588 + 0.837402i \(0.315926\pi\)
\(674\) 35.6124 1.37174
\(675\) 1.00000 0.0384900
\(676\) −72.8284 −2.80109
\(677\) 5.08697 0.195508 0.0977540 0.995211i \(-0.468834\pi\)
0.0977540 + 0.995211i \(0.468834\pi\)
\(678\) −0.393801 −0.0151238
\(679\) −19.6967 −0.755889
\(680\) −51.0813 −1.95888
\(681\) −8.28399 −0.317443
\(682\) 0 0
\(683\) 17.3612 0.664310 0.332155 0.943225i \(-0.392224\pi\)
0.332155 + 0.943225i \(0.392224\pi\)
\(684\) −26.6975 −1.02081
\(685\) −12.3448 −0.471670
\(686\) 55.6425 2.12444
\(687\) −2.87622 −0.109735
\(688\) −118.713 −4.52588
\(689\) −4.72279 −0.179924
\(690\) −2.34478 −0.0892641
\(691\) −42.9791 −1.63500 −0.817500 0.575928i \(-0.804641\pi\)
−0.817500 + 0.575928i \(0.804641\pi\)
\(692\) 38.3983 1.45969
\(693\) 0 0
\(694\) −5.23870 −0.198858
\(695\) −7.73236 −0.293305
\(696\) −27.1457 −1.02896
\(697\) −21.4709 −0.813267
\(698\) 65.9198 2.49510
\(699\) 5.17868 0.195876
\(700\) 12.9486 0.489411
\(701\) 14.3882 0.543436 0.271718 0.962377i \(-0.412408\pi\)
0.271718 + 0.962377i \(0.412408\pi\)
\(702\) −1.20796 −0.0455915
\(703\) 41.6174 1.56963
\(704\) 0 0
\(705\) −0.468179 −0.0176326
\(706\) 56.0805 2.11062
\(707\) 12.9396 0.486644
\(708\) 22.3769 0.840976
\(709\) 51.1751 1.92192 0.960961 0.276682i \(-0.0892349\pi\)
0.960961 + 0.276682i \(0.0892349\pi\)
\(710\) −31.3649 −1.17710
\(711\) 10.8707 0.407682
\(712\) −60.2716 −2.25877
\(713\) 3.94812 0.147858
\(714\) 31.5700 1.18148
\(715\) 0 0
\(716\) 37.0787 1.38570
\(717\) −10.8328 −0.404557
\(718\) −51.5349 −1.92327
\(719\) 52.0371 1.94066 0.970328 0.241793i \(-0.0777354\pi\)
0.970328 + 0.241793i \(0.0777354\pi\)
\(720\) −16.9513 −0.631737
\(721\) −2.95205 −0.109940
\(722\) −8.46084 −0.314880
\(723\) 2.96526 0.110279
\(724\) 78.1841 2.90569
\(725\) 2.65711 0.0986826
\(726\) 0 0
\(727\) −18.2951 −0.678528 −0.339264 0.940691i \(-0.610178\pi\)
−0.339264 + 0.940691i \(0.610178\pi\)
\(728\) −10.1389 −0.375773
\(729\) 1.00000 0.0370370
\(730\) −23.3673 −0.864864
\(731\) 35.0158 1.29511
\(732\) −16.8595 −0.623144
\(733\) −37.5153 −1.38566 −0.692830 0.721101i \(-0.743636\pi\)
−0.692830 + 0.721101i \(0.743636\pi\)
\(734\) −16.3786 −0.604545
\(735\) 1.81258 0.0668580
\(736\) 22.4649 0.828069
\(737\) 0 0
\(738\) −11.9044 −0.438207
\(739\) 27.3138 1.00475 0.502377 0.864649i \(-0.332459\pi\)
0.502377 + 0.864649i \(0.332459\pi\)
\(740\) 50.3847 1.85218
\(741\) −2.04620 −0.0751691
\(742\) 68.4349 2.51233
\(743\) −18.8234 −0.690562 −0.345281 0.938499i \(-0.612216\pi\)
−0.345281 + 0.938499i \(0.612216\pi\)
\(744\) 47.6879 1.74832
\(745\) 7.60895 0.278770
\(746\) −4.62938 −0.169494
\(747\) 5.92050 0.216620
\(748\) 0 0
\(749\) 27.7884 1.01536
\(750\) 2.77222 0.101227
\(751\) −25.1894 −0.919174 −0.459587 0.888133i \(-0.652003\pi\)
−0.459587 + 0.888133i \(0.652003\pi\)
\(752\) 7.93624 0.289405
\(753\) −17.6144 −0.641905
\(754\) −3.20968 −0.116890
\(755\) −2.67425 −0.0973259
\(756\) 12.9486 0.470936
\(757\) 45.7939 1.66441 0.832204 0.554470i \(-0.187079\pi\)
0.832204 + 0.554470i \(0.187079\pi\)
\(758\) −23.3842 −0.849352
\(759\) 0 0
\(760\) −47.9751 −1.74024
\(761\) −8.01793 −0.290650 −0.145325 0.989384i \(-0.546423\pi\)
−0.145325 + 0.989384i \(0.546423\pi\)
\(762\) −1.39640 −0.0505864
\(763\) 27.7055 1.00300
\(764\) −110.498 −3.99768
\(765\) 5.00000 0.180775
\(766\) −60.6605 −2.19175
\(767\) 1.71505 0.0619269
\(768\) 78.6025 2.83632
\(769\) −40.0439 −1.44402 −0.722010 0.691883i \(-0.756782\pi\)
−0.722010 + 0.691883i \(0.756782\pi\)
\(770\) 0 0
\(771\) 20.9647 0.755025
\(772\) −128.556 −4.62684
\(773\) −23.8250 −0.856925 −0.428462 0.903560i \(-0.640945\pi\)
−0.428462 + 0.903560i \(0.640945\pi\)
\(774\) 19.4143 0.697834
\(775\) −4.66785 −0.167674
\(776\) 88.3504 3.17160
\(777\) −20.1849 −0.724130
\(778\) 1.08697 0.0389697
\(779\) −20.1652 −0.722495
\(780\) −2.47726 −0.0887001
\(781\) 0 0
\(782\) −11.7239 −0.419245
\(783\) 2.65711 0.0949574
\(784\) −30.7255 −1.09734
\(785\) −20.2000 −0.720968
\(786\) 52.9766 1.88961
\(787\) 9.48879 0.338239 0.169119 0.985596i \(-0.445908\pi\)
0.169119 + 0.985596i \(0.445908\pi\)
\(788\) −130.931 −4.66421
\(789\) −8.43471 −0.300284
\(790\) 30.1360 1.07219
\(791\) 0.323537 0.0115037
\(792\) 0 0
\(793\) −1.29217 −0.0458864
\(794\) −99.1966 −3.52036
\(795\) 10.8386 0.384406
\(796\) −111.553 −3.95389
\(797\) −23.7639 −0.841759 −0.420880 0.907117i \(-0.638279\pi\)
−0.420880 + 0.907117i \(0.638279\pi\)
\(798\) 29.6502 1.04961
\(799\) −2.34090 −0.0828149
\(800\) −26.5602 −0.939046
\(801\) 5.89958 0.208451
\(802\) −84.3212 −2.97749
\(803\) 0 0
\(804\) −14.0452 −0.495337
\(805\) 1.92641 0.0678971
\(806\) 5.63857 0.198610
\(807\) 9.64346 0.339466
\(808\) −58.0412 −2.04188
\(809\) 18.4128 0.647361 0.323680 0.946167i \(-0.395080\pi\)
0.323680 + 0.946167i \(0.395080\pi\)
\(810\) 2.77222 0.0974060
\(811\) −23.6451 −0.830293 −0.415147 0.909755i \(-0.636270\pi\)
−0.415147 + 0.909755i \(0.636270\pi\)
\(812\) 34.4059 1.20741
\(813\) −10.2278 −0.358704
\(814\) 0 0
\(815\) 9.79582 0.343133
\(816\) −84.7564 −2.96707
\(817\) 32.8866 1.15056
\(818\) −45.1033 −1.57700
\(819\) 0.992430 0.0346783
\(820\) −24.4133 −0.852550
\(821\) −26.3153 −0.918409 −0.459204 0.888331i \(-0.651865\pi\)
−0.459204 + 0.888331i \(0.651865\pi\)
\(822\) −34.2225 −1.19365
\(823\) 40.4318 1.40936 0.704681 0.709524i \(-0.251090\pi\)
0.704681 + 0.709524i \(0.251090\pi\)
\(824\) 13.2416 0.461293
\(825\) 0 0
\(826\) −24.8517 −0.864703
\(827\) −16.5927 −0.576984 −0.288492 0.957482i \(-0.593154\pi\)
−0.288492 + 0.957482i \(0.593154\pi\)
\(828\) −4.80862 −0.167111
\(829\) −34.3073 −1.19154 −0.595770 0.803155i \(-0.703153\pi\)
−0.595770 + 0.803155i \(0.703153\pi\)
\(830\) 16.4129 0.569701
\(831\) 5.83903 0.202554
\(832\) 17.3111 0.600154
\(833\) 9.06289 0.314011
\(834\) −21.4358 −0.742261
\(835\) 25.5776 0.885150
\(836\) 0 0
\(837\) −4.66785 −0.161344
\(838\) 111.084 3.83732
\(839\) −3.38811 −0.116971 −0.0584853 0.998288i \(-0.518627\pi\)
−0.0584853 + 0.998288i \(0.518627\pi\)
\(840\) 23.2684 0.802838
\(841\) −21.9398 −0.756543
\(842\) 53.7652 1.85287
\(843\) −14.4139 −0.496439
\(844\) −0.283836 −0.00977003
\(845\) 12.8101 0.440682
\(846\) −1.29790 −0.0446226
\(847\) 0 0
\(848\) −183.729 −6.30926
\(849\) 10.6334 0.364939
\(850\) 13.8611 0.475432
\(851\) 7.49591 0.256956
\(852\) −64.3225 −2.20365
\(853\) −16.4060 −0.561732 −0.280866 0.959747i \(-0.590622\pi\)
−0.280866 + 0.959747i \(0.590622\pi\)
\(854\) 18.7241 0.640725
\(855\) 4.69596 0.160598
\(856\) −124.646 −4.26032
\(857\) 3.37817 0.115396 0.0576981 0.998334i \(-0.481624\pi\)
0.0576981 + 0.998334i \(0.481624\pi\)
\(858\) 0 0
\(859\) −2.32376 −0.0792855 −0.0396428 0.999214i \(-0.512622\pi\)
−0.0396428 + 0.999214i \(0.512622\pi\)
\(860\) 39.8145 1.35766
\(861\) 9.78037 0.333314
\(862\) 94.1015 3.20511
\(863\) −20.2652 −0.689837 −0.344918 0.938633i \(-0.612093\pi\)
−0.344918 + 0.938633i \(0.612093\pi\)
\(864\) −26.5602 −0.903598
\(865\) −6.75406 −0.229645
\(866\) 36.3591 1.23553
\(867\) 8.00000 0.271694
\(868\) −60.4421 −2.05154
\(869\) 0 0
\(870\) 7.36611 0.249734
\(871\) −1.07648 −0.0364751
\(872\) −124.274 −4.20846
\(873\) −8.64803 −0.292691
\(874\) −11.0110 −0.372452
\(875\) −2.27759 −0.0769966
\(876\) −47.9213 −1.61911
\(877\) 16.6208 0.561244 0.280622 0.959818i \(-0.409459\pi\)
0.280622 + 0.959818i \(0.409459\pi\)
\(878\) −26.7445 −0.902583
\(879\) 8.39576 0.283182
\(880\) 0 0
\(881\) −29.8895 −1.00700 −0.503502 0.863994i \(-0.667955\pi\)
−0.503502 + 0.863994i \(0.667955\pi\)
\(882\) 5.02487 0.169196
\(883\) 14.0135 0.471593 0.235796 0.971803i \(-0.424230\pi\)
0.235796 + 0.971803i \(0.424230\pi\)
\(884\) −12.3863 −0.416596
\(885\) −3.93598 −0.132307
\(886\) 24.7119 0.830211
\(887\) −28.6081 −0.960567 −0.480283 0.877113i \(-0.659466\pi\)
−0.480283 + 0.877113i \(0.659466\pi\)
\(888\) 90.5404 3.03834
\(889\) 1.14725 0.0384776
\(890\) 16.3550 0.548219
\(891\) 0 0
\(892\) 4.29213 0.143711
\(893\) −2.19855 −0.0735716
\(894\) 21.0937 0.705479
\(895\) −6.52195 −0.218005
\(896\) −129.858 −4.33824
\(897\) −0.368551 −0.0123056
\(898\) 34.4777 1.15054
\(899\) −12.4030 −0.413663
\(900\) 5.68522 0.189507
\(901\) 54.1931 1.80543
\(902\) 0 0
\(903\) −15.9504 −0.530794
\(904\) −1.45124 −0.0482676
\(905\) −13.7522 −0.457138
\(906\) −7.41362 −0.246301
\(907\) 26.9693 0.895499 0.447750 0.894159i \(-0.352226\pi\)
0.447750 + 0.894159i \(0.352226\pi\)
\(908\) −47.0963 −1.56295
\(909\) 5.68126 0.188436
\(910\) 2.75124 0.0912027
\(911\) 7.13660 0.236446 0.118223 0.992987i \(-0.462280\pi\)
0.118223 + 0.992987i \(0.462280\pi\)
\(912\) −79.6025 −2.63590
\(913\) 0 0
\(914\) −105.783 −3.49900
\(915\) 2.96549 0.0980361
\(916\) −16.3519 −0.540284
\(917\) −43.5243 −1.43730
\(918\) 13.8611 0.457485
\(919\) 6.89424 0.227420 0.113710 0.993514i \(-0.463727\pi\)
0.113710 + 0.993514i \(0.463727\pi\)
\(920\) −8.64102 −0.284886
\(921\) −4.51902 −0.148907
\(922\) 95.3219 3.13926
\(923\) −4.92992 −0.162270
\(924\) 0 0
\(925\) −8.86239 −0.291394
\(926\) 111.599 3.66737
\(927\) −1.29613 −0.0425705
\(928\) −70.5735 −2.31669
\(929\) −36.3816 −1.19364 −0.596821 0.802375i \(-0.703570\pi\)
−0.596821 + 0.802375i \(0.703570\pi\)
\(930\) −12.9403 −0.424330
\(931\) 8.51179 0.278963
\(932\) 29.4420 0.964403
\(933\) 23.9970 0.785626
\(934\) −41.9052 −1.37118
\(935\) 0 0
\(936\) −4.45160 −0.145505
\(937\) −52.6934 −1.72142 −0.860710 0.509096i \(-0.829980\pi\)
−0.860710 + 0.509096i \(0.829980\pi\)
\(938\) 15.5986 0.509312
\(939\) −25.8686 −0.844190
\(940\) −2.66170 −0.0868151
\(941\) 20.2984 0.661709 0.330854 0.943682i \(-0.392663\pi\)
0.330854 + 0.943682i \(0.392663\pi\)
\(942\) −55.9988 −1.82454
\(943\) −3.63206 −0.118276
\(944\) 66.7199 2.17155
\(945\) −2.27759 −0.0740900
\(946\) 0 0
\(947\) 14.5680 0.473395 0.236698 0.971583i \(-0.423935\pi\)
0.236698 + 0.971583i \(0.423935\pi\)
\(948\) 61.8022 2.00724
\(949\) −3.67287 −0.119226
\(950\) 13.0182 0.422367
\(951\) 2.90319 0.0941425
\(952\) 116.342 3.77067
\(953\) 39.5054 1.27970 0.639852 0.768498i \(-0.278996\pi\)
0.639852 + 0.768498i \(0.278996\pi\)
\(954\) 30.0471 0.972810
\(955\) 19.4360 0.628936
\(956\) −61.5867 −1.99186
\(957\) 0 0
\(958\) −79.8524 −2.57991
\(959\) 28.1164 0.907924
\(960\) −39.7283 −1.28223
\(961\) −9.21120 −0.297135
\(962\) 10.7054 0.345156
\(963\) 12.2008 0.393164
\(964\) 16.8581 0.542964
\(965\) 22.6124 0.727917
\(966\) 5.34044 0.171826
\(967\) 44.8051 1.44083 0.720417 0.693541i \(-0.243950\pi\)
0.720417 + 0.693541i \(0.243950\pi\)
\(968\) 0 0
\(969\) 23.4798 0.754279
\(970\) −23.9743 −0.769767
\(971\) 18.1454 0.582313 0.291156 0.956675i \(-0.405960\pi\)
0.291156 + 0.956675i \(0.405960\pi\)
\(972\) 5.68522 0.182354
\(973\) 17.6111 0.564587
\(974\) −79.9856 −2.56290
\(975\) 0.435737 0.0139547
\(976\) −50.2689 −1.60907
\(977\) 51.9967 1.66352 0.831761 0.555134i \(-0.187333\pi\)
0.831761 + 0.555134i \(0.187333\pi\)
\(978\) 27.1562 0.868359
\(979\) 0 0
\(980\) 10.3049 0.329178
\(981\) 12.1644 0.388378
\(982\) −41.4079 −1.32138
\(983\) 44.3322 1.41398 0.706989 0.707224i \(-0.250053\pi\)
0.706989 + 0.707224i \(0.250053\pi\)
\(984\) −43.8703 −1.39854
\(985\) 23.0300 0.733797
\(986\) 36.8305 1.17292
\(987\) 1.06632 0.0339414
\(988\) −11.6331 −0.370098
\(989\) 5.92336 0.188352
\(990\) 0 0
\(991\) 10.4084 0.330635 0.165317 0.986240i \(-0.447135\pi\)
0.165317 + 0.986240i \(0.447135\pi\)
\(992\) 123.979 3.93634
\(993\) 10.9837 0.348558
\(994\) 71.4364 2.26583
\(995\) 19.6216 0.622046
\(996\) 33.6593 1.06654
\(997\) −6.52202 −0.206554 −0.103277 0.994653i \(-0.532933\pi\)
−0.103277 + 0.994653i \(0.532933\pi\)
\(998\) −61.5406 −1.94803
\(999\) −8.86239 −0.280394
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 1815.2.a.o.1.1 4
3.2 odd 2 5445.2.a.bv.1.4 4
5.4 even 2 9075.2.a.dj.1.4 4
11.2 odd 10 165.2.m.a.136.1 yes 8
11.6 odd 10 165.2.m.a.91.1 8
11.10 odd 2 1815.2.a.x.1.4 4
33.2 even 10 495.2.n.d.136.2 8
33.17 even 10 495.2.n.d.91.2 8
33.32 even 2 5445.2.a.be.1.1 4
55.2 even 20 825.2.bx.h.499.4 16
55.13 even 20 825.2.bx.h.499.1 16
55.17 even 20 825.2.bx.h.124.1 16
55.24 odd 10 825.2.n.k.301.2 8
55.28 even 20 825.2.bx.h.124.4 16
55.39 odd 10 825.2.n.k.751.2 8
55.54 odd 2 9075.2.a.cl.1.1 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.91.1 8 11.6 odd 10
165.2.m.a.136.1 yes 8 11.2 odd 10
495.2.n.d.91.2 8 33.17 even 10
495.2.n.d.136.2 8 33.2 even 10
825.2.n.k.301.2 8 55.24 odd 10
825.2.n.k.751.2 8 55.39 odd 10
825.2.bx.h.124.1 16 55.17 even 20
825.2.bx.h.124.4 16 55.28 even 20
825.2.bx.h.499.1 16 55.13 even 20
825.2.bx.h.499.4 16 55.2 even 20
1815.2.a.o.1.1 4 1.1 even 1 trivial
1815.2.a.x.1.4 4 11.10 odd 2
5445.2.a.be.1.1 4 33.32 even 2
5445.2.a.bv.1.4 4 3.2 odd 2
9075.2.a.cl.1.1 4 55.54 odd 2
9075.2.a.dj.1.4 4 5.4 even 2