Properties

Label 1815.2.a.o.1.3
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 1.3
Root \(-1.35567\) of defining polynomial
Character \(\chi\) \(=\) 1815.1

$q$-expansion

\(f(q)\) \(=\) \(q-1.16215 q^{2} +1.00000 q^{3} -0.649414 q^{4} -1.00000 q^{5} -1.16215 q^{6} +4.28684 q^{7} +3.07901 q^{8} +1.00000 q^{9} +O(q^{10})\) \(q-1.16215 q^{2} +1.00000 q^{3} -0.649414 q^{4} -1.00000 q^{5} -1.16215 q^{6} +4.28684 q^{7} +3.07901 q^{8} +1.00000 q^{9} +1.16215 q^{10} -0.649414 q^{12} -5.16724 q^{13} -4.98194 q^{14} -1.00000 q^{15} -2.27943 q^{16} -5.00000 q^{17} -1.16215 q^{18} -5.59998 q^{19} +0.649414 q^{20} +4.28684 q^{21} -0.219819 q^{23} +3.07901 q^{24} +1.00000 q^{25} +6.00509 q^{26} +1.00000 q^{27} -2.78393 q^{28} -6.41843 q^{29} +1.16215 q^{30} -2.83095 q^{31} -3.50898 q^{32} +5.81074 q^{34} -4.28684 q^{35} -0.649414 q^{36} +3.92802 q^{37} +6.50800 q^{38} -5.16724 q^{39} -3.07901 q^{40} -5.86100 q^{41} -4.98194 q^{42} +8.90173 q^{43} -1.00000 q^{45} +0.255462 q^{46} -0.237878 q^{47} -2.27943 q^{48} +11.3770 q^{49} -1.16215 q^{50} -5.00000 q^{51} +3.35567 q^{52} +2.53671 q^{53} -1.16215 q^{54} +13.1992 q^{56} -5.59998 q^{57} +7.45917 q^{58} -7.87035 q^{59} +0.649414 q^{60} +4.85807 q^{61} +3.28999 q^{62} +4.28684 q^{63} +8.63682 q^{64} +5.16724 q^{65} +12.1280 q^{67} +3.24707 q^{68} -0.219819 q^{69} +4.98194 q^{70} -9.98194 q^{71} +3.07901 q^{72} -14.5625 q^{73} -4.56494 q^{74} +1.00000 q^{75} +3.63670 q^{76} +6.00509 q^{78} -8.00194 q^{79} +2.27943 q^{80} +1.00000 q^{81} +6.81134 q^{82} -12.5530 q^{83} -2.78393 q^{84} +5.00000 q^{85} -10.3451 q^{86} -6.41843 q^{87} -2.56545 q^{89} +1.16215 q^{90} -22.1511 q^{91} +0.142753 q^{92} -2.83095 q^{93} +0.276449 q^{94} +5.59998 q^{95} -3.50898 q^{96} +2.01199 q^{97} -13.2218 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q - 5q^{2} + 4q^{3} + 9q^{4} - 4q^{5} - 5q^{6} + 2q^{7} - 15q^{8} + 4q^{9} + O(q^{10}) \) \( 4q - 5q^{2} + 4q^{3} + 9q^{4} - 4q^{5} - 5q^{6} + 2q^{7} - 15q^{8} + 4q^{9} + 5q^{10} + 9q^{12} - 3q^{13} - 5q^{14} - 4q^{15} + 15q^{16} - 20q^{17} - 5q^{18} - 3q^{19} - 9q^{20} + 2q^{21} - 5q^{23} - 15q^{24} + 4q^{25} + 6q^{26} + 4q^{27} - 3q^{28} - 5q^{29} + 5q^{30} - q^{31} - 30q^{32} + 25q^{34} - 2q^{35} + 9q^{36} - 7q^{37} + q^{38} - 3q^{39} + 15q^{40} - 20q^{41} - 5q^{42} + 2q^{43} - 4q^{45} - 7q^{46} - 20q^{47} + 15q^{48} + 8q^{49} - 5q^{50} - 20q^{51} + 7q^{52} + 6q^{53} - 5q^{54} + 10q^{56} - 3q^{57} - 21q^{58} - 5q^{59} - 9q^{60} + 7q^{61} + 12q^{62} + 2q^{63} + 49q^{64} + 3q^{65} - 13q^{67} - 45q^{68} - 5q^{69} + 5q^{70} - 25q^{71} - 15q^{72} - 23q^{73} + 7q^{74} + 4q^{75} + 7q^{76} + 6q^{78} - 15q^{80} + 4q^{81} + 11q^{82} - 33q^{83} - 3q^{84} + 20q^{85} - 12q^{86} - 5q^{87} + 16q^{89} + 5q^{90} - 24q^{91} - q^{93} + 17q^{94} + 3q^{95} - 30q^{96} - 25q^{98} + O(q^{100}) \)

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:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) −1.16215 −0.821762 −0.410881 0.911689i \(-0.634779\pi\)
−0.410881 + 0.911689i \(0.634779\pi\)
\(3\) 1.00000 0.577350
\(4\) −0.649414 −0.324707
\(5\) −1.00000 −0.447214
\(6\) −1.16215 −0.474445
\(7\) 4.28684 1.62027 0.810137 0.586241i \(-0.199393\pi\)
0.810137 + 0.586241i \(0.199393\pi\)
\(8\) 3.07901 1.08859
\(9\) 1.00000 0.333333
\(10\) 1.16215 0.367503
\(11\) 0 0
\(12\) −0.649414 −0.187470
\(13\) −5.16724 −1.43313 −0.716567 0.697519i \(-0.754287\pi\)
−0.716567 + 0.697519i \(0.754287\pi\)
\(14\) −4.98194 −1.33148
\(15\) −1.00000 −0.258199
\(16\) −2.27943 −0.569858
\(17\) −5.00000 −1.21268 −0.606339 0.795206i \(-0.707363\pi\)
−0.606339 + 0.795206i \(0.707363\pi\)
\(18\) −1.16215 −0.273921
\(19\) −5.59998 −1.28472 −0.642361 0.766402i \(-0.722045\pi\)
−0.642361 + 0.766402i \(0.722045\pi\)
\(20\) 0.649414 0.145213
\(21\) 4.28684 0.935466
\(22\) 0 0
\(23\) −0.219819 −0.0458354 −0.0229177 0.999737i \(-0.507296\pi\)
−0.0229177 + 0.999737i \(0.507296\pi\)
\(24\) 3.07901 0.628500
\(25\) 1.00000 0.200000
\(26\) 6.00509 1.17769
\(27\) 1.00000 0.192450
\(28\) −2.78393 −0.526114
\(29\) −6.41843 −1.19187 −0.595937 0.803031i \(-0.703219\pi\)
−0.595937 + 0.803031i \(0.703219\pi\)
\(30\) 1.16215 0.212178
\(31\) −2.83095 −0.508455 −0.254227 0.967145i \(-0.581821\pi\)
−0.254227 + 0.967145i \(0.581821\pi\)
\(32\) −3.50898 −0.620306
\(33\) 0 0
\(34\) 5.81074 0.996533
\(35\) −4.28684 −0.724608
\(36\) −0.649414 −0.108236
\(37\) 3.92802 0.645763 0.322881 0.946439i \(-0.395348\pi\)
0.322881 + 0.946439i \(0.395348\pi\)
\(38\) 6.50800 1.05574
\(39\) −5.16724 −0.827420
\(40\) −3.07901 −0.486834
\(41\) −5.86100 −0.915334 −0.457667 0.889124i \(-0.651315\pi\)
−0.457667 + 0.889124i \(0.651315\pi\)
\(42\) −4.98194 −0.768730
\(43\) 8.90173 1.35750 0.678751 0.734369i \(-0.262522\pi\)
0.678751 + 0.734369i \(0.262522\pi\)
\(44\) 0 0
\(45\) −1.00000 −0.149071
\(46\) 0.255462 0.0376658
\(47\) −0.237878 −0.0346980 −0.0173490 0.999849i \(-0.505523\pi\)
−0.0173490 + 0.999849i \(0.505523\pi\)
\(48\) −2.27943 −0.329008
\(49\) 11.3770 1.62529
\(50\) −1.16215 −0.164352
\(51\) −5.00000 −0.700140
\(52\) 3.35567 0.465348
\(53\) 2.53671 0.348443 0.174222 0.984706i \(-0.444259\pi\)
0.174222 + 0.984706i \(0.444259\pi\)
\(54\) −1.16215 −0.158148
\(55\) 0 0
\(56\) 13.1992 1.76382
\(57\) −5.59998 −0.741735
\(58\) 7.45917 0.979436
\(59\) −7.87035 −1.02463 −0.512316 0.858797i \(-0.671212\pi\)
−0.512316 + 0.858797i \(0.671212\pi\)
\(60\) 0.649414 0.0838390
\(61\) 4.85807 0.622012 0.311006 0.950408i \(-0.399334\pi\)
0.311006 + 0.950408i \(0.399334\pi\)
\(62\) 3.28999 0.417829
\(63\) 4.28684 0.540091
\(64\) 8.63682 1.07960
\(65\) 5.16724 0.640917
\(66\) 0 0
\(67\) 12.1280 1.48167 0.740834 0.671688i \(-0.234431\pi\)
0.740834 + 0.671688i \(0.234431\pi\)
\(68\) 3.24707 0.393765
\(69\) −0.219819 −0.0264631
\(70\) 4.98194 0.595456
\(71\) −9.98194 −1.18464 −0.592319 0.805703i \(-0.701788\pi\)
−0.592319 + 0.805703i \(0.701788\pi\)
\(72\) 3.07901 0.362865
\(73\) −14.5625 −1.70441 −0.852207 0.523205i \(-0.824736\pi\)
−0.852207 + 0.523205i \(0.824736\pi\)
\(74\) −4.56494 −0.530664
\(75\) 1.00000 0.115470
\(76\) 3.63670 0.417158
\(77\) 0 0
\(78\) 6.00509 0.679942
\(79\) −8.00194 −0.900289 −0.450144 0.892956i \(-0.648628\pi\)
−0.450144 + 0.892956i \(0.648628\pi\)
\(80\) 2.27943 0.254848
\(81\) 1.00000 0.111111
\(82\) 6.81134 0.752187
\(83\) −12.5530 −1.37787 −0.688933 0.724825i \(-0.741921\pi\)
−0.688933 + 0.724825i \(0.741921\pi\)
\(84\) −2.78393 −0.303752
\(85\) 5.00000 0.542326
\(86\) −10.3451 −1.11554
\(87\) −6.41843 −0.688128
\(88\) 0 0
\(89\) −2.56545 −0.271937 −0.135969 0.990713i \(-0.543415\pi\)
−0.135969 + 0.990713i \(0.543415\pi\)
\(90\) 1.16215 0.122501
\(91\) −22.1511 −2.32207
\(92\) 0.142753 0.0148831
\(93\) −2.83095 −0.293556
\(94\) 0.276449 0.0285135
\(95\) 5.59998 0.574545
\(96\) −3.50898 −0.358134
\(97\) 2.01199 0.204286 0.102143 0.994770i \(-0.467430\pi\)
0.102143 + 0.994770i \(0.467430\pi\)
\(98\) −13.2218 −1.33560
\(99\) 0 0
\(100\) −0.649414 −0.0649414
\(101\) 7.58484 0.754720 0.377360 0.926067i \(-0.376832\pi\)
0.377360 + 0.926067i \(0.376832\pi\)
\(102\) 5.81074 0.575349
\(103\) −16.8685 −1.66211 −0.831053 0.556193i \(-0.812262\pi\)
−0.831053 + 0.556193i \(0.812262\pi\)
\(104\) −15.9100 −1.56010
\(105\) −4.28684 −0.418353
\(106\) −2.94803 −0.286338
\(107\) −7.76161 −0.750343 −0.375172 0.926955i \(-0.622416\pi\)
−0.375172 + 0.926955i \(0.622416\pi\)
\(108\) −0.649414 −0.0624899
\(109\) −4.45671 −0.426876 −0.213438 0.976957i \(-0.568466\pi\)
−0.213438 + 0.976957i \(0.568466\pi\)
\(110\) 0 0
\(111\) 3.92802 0.372831
\(112\) −9.77157 −0.923327
\(113\) −7.71247 −0.725528 −0.362764 0.931881i \(-0.618167\pi\)
−0.362764 + 0.931881i \(0.618167\pi\)
\(114\) 6.50800 0.609530
\(115\) 0.219819 0.0204982
\(116\) 4.16822 0.387010
\(117\) −5.16724 −0.477711
\(118\) 9.14651 0.842004
\(119\) −21.4342 −1.96487
\(120\) −3.07901 −0.281074
\(121\) 0 0
\(122\) −5.64579 −0.511146
\(123\) −5.86100 −0.528469
\(124\) 1.83846 0.165099
\(125\) −1.00000 −0.0894427
\(126\) −4.98194 −0.443827
\(127\) −1.53114 −0.135867 −0.0679335 0.997690i \(-0.521641\pi\)
−0.0679335 + 0.997690i \(0.521641\pi\)
\(128\) −3.01930 −0.266871
\(129\) 8.90173 0.783754
\(130\) −6.00509 −0.526681
\(131\) −2.66108 −0.232500 −0.116250 0.993220i \(-0.537087\pi\)
−0.116250 + 0.993220i \(0.537087\pi\)
\(132\) 0 0
\(133\) −24.0062 −2.08160
\(134\) −14.0945 −1.21758
\(135\) −1.00000 −0.0860663
\(136\) −15.3950 −1.32011
\(137\) 10.2555 0.876183 0.438092 0.898930i \(-0.355655\pi\)
0.438092 + 0.898930i \(0.355655\pi\)
\(138\) 0.255462 0.0217464
\(139\) 5.29507 0.449122 0.224561 0.974460i \(-0.427905\pi\)
0.224561 + 0.974460i \(0.427905\pi\)
\(140\) 2.78393 0.235285
\(141\) −0.237878 −0.0200329
\(142\) 11.6005 0.973491
\(143\) 0 0
\(144\) −2.27943 −0.189953
\(145\) 6.41843 0.533022
\(146\) 16.9238 1.40062
\(147\) 11.3770 0.938360
\(148\) −2.55091 −0.209684
\(149\) −3.78841 −0.310359 −0.155179 0.987886i \(-0.549596\pi\)
−0.155179 + 0.987886i \(0.549596\pi\)
\(150\) −1.16215 −0.0948889
\(151\) 5.92001 0.481763 0.240882 0.970554i \(-0.422563\pi\)
0.240882 + 0.970554i \(0.422563\pi\)
\(152\) −17.2424 −1.39854
\(153\) −5.00000 −0.404226
\(154\) 0 0
\(155\) 2.83095 0.227388
\(156\) 3.35567 0.268669
\(157\) −7.42909 −0.592906 −0.296453 0.955048i \(-0.595804\pi\)
−0.296453 + 0.955048i \(0.595804\pi\)
\(158\) 9.29944 0.739823
\(159\) 2.53671 0.201174
\(160\) 3.50898 0.277409
\(161\) −0.942328 −0.0742659
\(162\) −1.16215 −0.0913069
\(163\) 5.32086 0.416762 0.208381 0.978048i \(-0.433181\pi\)
0.208381 + 0.978048i \(0.433181\pi\)
\(164\) 3.80621 0.297215
\(165\) 0 0
\(166\) 14.5884 1.13228
\(167\) 1.97139 0.152551 0.0762753 0.997087i \(-0.475697\pi\)
0.0762753 + 0.997087i \(0.475697\pi\)
\(168\) 13.1992 1.01834
\(169\) 13.7003 1.05387
\(170\) −5.81074 −0.445663
\(171\) −5.59998 −0.428241
\(172\) −5.78091 −0.440790
\(173\) 1.97504 0.150160 0.0750799 0.997178i \(-0.476079\pi\)
0.0750799 + 0.997178i \(0.476079\pi\)
\(174\) 7.45917 0.565478
\(175\) 4.28684 0.324055
\(176\) 0 0
\(177\) −7.87035 −0.591572
\(178\) 2.98143 0.223468
\(179\) −2.02315 −0.151217 −0.0756086 0.997138i \(-0.524090\pi\)
−0.0756086 + 0.997138i \(0.524090\pi\)
\(180\) 0.649414 0.0484045
\(181\) 20.1380 1.49685 0.748423 0.663221i \(-0.230811\pi\)
0.748423 + 0.663221i \(0.230811\pi\)
\(182\) 25.7429 1.90819
\(183\) 4.85807 0.359119
\(184\) −0.676824 −0.0498961
\(185\) −3.92802 −0.288794
\(186\) 3.28999 0.241234
\(187\) 0 0
\(188\) 0.154481 0.0112667
\(189\) 4.28684 0.311822
\(190\) −6.50800 −0.472140
\(191\) 12.6652 0.916418 0.458209 0.888844i \(-0.348491\pi\)
0.458209 + 0.888844i \(0.348491\pi\)
\(192\) 8.63682 0.623309
\(193\) 13.9321 1.00286 0.501428 0.865199i \(-0.332808\pi\)
0.501428 + 0.865199i \(0.332808\pi\)
\(194\) −2.33822 −0.167875
\(195\) 5.16724 0.370033
\(196\) −7.38839 −0.527742
\(197\) −14.6060 −1.04064 −0.520319 0.853972i \(-0.674187\pi\)
−0.520319 + 0.853972i \(0.674187\pi\)
\(198\) 0 0
\(199\) −11.8748 −0.841784 −0.420892 0.907111i \(-0.638283\pi\)
−0.420892 + 0.907111i \(0.638283\pi\)
\(200\) 3.07901 0.217719
\(201\) 12.1280 0.855441
\(202\) −8.81471 −0.620201
\(203\) −27.5148 −1.93116
\(204\) 3.24707 0.227340
\(205\) 5.86100 0.409350
\(206\) 19.6037 1.36586
\(207\) −0.219819 −0.0152785
\(208\) 11.7784 0.816683
\(209\) 0 0
\(210\) 4.98194 0.343787
\(211\) 22.3518 1.53876 0.769382 0.638789i \(-0.220564\pi\)
0.769382 + 0.638789i \(0.220564\pi\)
\(212\) −1.64737 −0.113142
\(213\) −9.98194 −0.683951
\(214\) 9.02014 0.616604
\(215\) −8.90173 −0.607093
\(216\) 3.07901 0.209500
\(217\) −12.1359 −0.823836
\(218\) 5.17936 0.350790
\(219\) −14.5625 −0.984044
\(220\) 0 0
\(221\) 25.8362 1.73793
\(222\) −4.56494 −0.306379
\(223\) −19.1583 −1.28294 −0.641468 0.767150i \(-0.721674\pi\)
−0.641468 + 0.767150i \(0.721674\pi\)
\(224\) −15.0424 −1.00507
\(225\) 1.00000 0.0666667
\(226\) 8.96302 0.596211
\(227\) −15.3759 −1.02053 −0.510267 0.860016i \(-0.670453\pi\)
−0.510267 + 0.860016i \(0.670453\pi\)
\(228\) 3.63670 0.240846
\(229\) 8.90126 0.588212 0.294106 0.955773i \(-0.404978\pi\)
0.294106 + 0.955773i \(0.404978\pi\)
\(230\) −0.255462 −0.0168447
\(231\) 0 0
\(232\) −19.7624 −1.29747
\(233\) −22.8362 −1.49605 −0.748024 0.663672i \(-0.768997\pi\)
−0.748024 + 0.663672i \(0.768997\pi\)
\(234\) 6.00509 0.392565
\(235\) 0.237878 0.0155174
\(236\) 5.11112 0.332705
\(237\) −8.00194 −0.519782
\(238\) 24.9097 1.61466
\(239\) −16.8605 −1.09062 −0.545308 0.838236i \(-0.683587\pi\)
−0.545308 + 0.838236i \(0.683587\pi\)
\(240\) 2.27943 0.147137
\(241\) 11.9607 0.770459 0.385229 0.922821i \(-0.374122\pi\)
0.385229 + 0.922821i \(0.374122\pi\)
\(242\) 0 0
\(243\) 1.00000 0.0641500
\(244\) −3.15490 −0.201972
\(245\) −11.3770 −0.726851
\(246\) 6.81134 0.434275
\(247\) 28.9364 1.84118
\(248\) −8.71654 −0.553501
\(249\) −12.5530 −0.795511
\(250\) 1.16215 0.0735006
\(251\) 16.0034 1.01013 0.505063 0.863082i \(-0.331469\pi\)
0.505063 + 0.863082i \(0.331469\pi\)
\(252\) −2.78393 −0.175371
\(253\) 0 0
\(254\) 1.77941 0.111650
\(255\) 5.00000 0.313112
\(256\) −13.7648 −0.860298
\(257\) 5.47446 0.341487 0.170744 0.985315i \(-0.445383\pi\)
0.170744 + 0.985315i \(0.445383\pi\)
\(258\) −10.3451 −0.644059
\(259\) 16.8388 1.04631
\(260\) −3.35567 −0.208110
\(261\) −6.41843 −0.397291
\(262\) 3.09257 0.191060
\(263\) −11.9841 −0.738971 −0.369486 0.929236i \(-0.620466\pi\)
−0.369486 + 0.929236i \(0.620466\pi\)
\(264\) 0 0
\(265\) −2.53671 −0.155829
\(266\) 27.8987 1.71058
\(267\) −2.56545 −0.157003
\(268\) −7.87607 −0.481108
\(269\) 13.6465 0.832041 0.416020 0.909355i \(-0.363424\pi\)
0.416020 + 0.909355i \(0.363424\pi\)
\(270\) 1.16215 0.0707260
\(271\) −11.8379 −0.719098 −0.359549 0.933126i \(-0.617070\pi\)
−0.359549 + 0.933126i \(0.617070\pi\)
\(272\) 11.3972 0.691055
\(273\) −22.1511 −1.34065
\(274\) −11.9184 −0.720014
\(275\) 0 0
\(276\) 0.142753 0.00859274
\(277\) −10.2638 −0.616694 −0.308347 0.951274i \(-0.599776\pi\)
−0.308347 + 0.951274i \(0.599776\pi\)
\(278\) −6.15366 −0.369072
\(279\) −2.83095 −0.169485
\(280\) −13.1992 −0.788804
\(281\) 2.93889 0.175319 0.0876597 0.996150i \(-0.472061\pi\)
0.0876597 + 0.996150i \(0.472061\pi\)
\(282\) 0.276449 0.0164623
\(283\) −21.8279 −1.29754 −0.648768 0.760986i \(-0.724716\pi\)
−0.648768 + 0.760986i \(0.724716\pi\)
\(284\) 6.48241 0.384660
\(285\) 5.59998 0.331714
\(286\) 0 0
\(287\) −25.1252 −1.48309
\(288\) −3.50898 −0.206769
\(289\) 8.00000 0.470588
\(290\) −7.45917 −0.438017
\(291\) 2.01199 0.117945
\(292\) 9.45710 0.553435
\(293\) 24.7665 1.44688 0.723438 0.690390i \(-0.242561\pi\)
0.723438 + 0.690390i \(0.242561\pi\)
\(294\) −13.2218 −0.771109
\(295\) 7.87035 0.458230
\(296\) 12.0944 0.702974
\(297\) 0 0
\(298\) 4.40269 0.255041
\(299\) 1.13586 0.0656882
\(300\) −0.649414 −0.0374939
\(301\) 38.1603 2.19952
\(302\) −6.87992 −0.395895
\(303\) 7.58484 0.435738
\(304\) 12.7648 0.732110
\(305\) −4.85807 −0.278172
\(306\) 5.81074 0.332178
\(307\) 4.94023 0.281954 0.140977 0.990013i \(-0.454976\pi\)
0.140977 + 0.990013i \(0.454976\pi\)
\(308\) 0 0
\(309\) −16.8685 −0.959618
\(310\) −3.28999 −0.186859
\(311\) −13.8096 −0.783070 −0.391535 0.920163i \(-0.628056\pi\)
−0.391535 + 0.920163i \(0.628056\pi\)
\(312\) −15.9100 −0.900724
\(313\) −4.06934 −0.230013 −0.115006 0.993365i \(-0.536689\pi\)
−0.115006 + 0.993365i \(0.536689\pi\)
\(314\) 8.63369 0.487227
\(315\) −4.28684 −0.241536
\(316\) 5.19657 0.292330
\(317\) 25.8199 1.45019 0.725096 0.688648i \(-0.241795\pi\)
0.725096 + 0.688648i \(0.241795\pi\)
\(318\) −2.94803 −0.165317
\(319\) 0 0
\(320\) −8.63682 −0.482813
\(321\) −7.76161 −0.433211
\(322\) 1.09512 0.0610289
\(323\) 27.9999 1.55795
\(324\) −0.649414 −0.0360785
\(325\) −5.16724 −0.286627
\(326\) −6.18362 −0.342479
\(327\) −4.45671 −0.246457
\(328\) −18.0461 −0.996428
\(329\) −1.01974 −0.0562203
\(330\) 0 0
\(331\) −3.35008 −0.184137 −0.0920684 0.995753i \(-0.529348\pi\)
−0.0920684 + 0.995753i \(0.529348\pi\)
\(332\) 8.15206 0.447403
\(333\) 3.92802 0.215254
\(334\) −2.29104 −0.125360
\(335\) −12.1280 −0.662622
\(336\) −9.77157 −0.533083
\(337\) 27.3998 1.49256 0.746282 0.665630i \(-0.231837\pi\)
0.746282 + 0.665630i \(0.231837\pi\)
\(338\) −15.9218 −0.866031
\(339\) −7.71247 −0.418884
\(340\) −3.24707 −0.176097
\(341\) 0 0
\(342\) 6.50800 0.351912
\(343\) 18.7636 1.01314
\(344\) 27.4085 1.47777
\(345\) 0.219819 0.0118346
\(346\) −2.29529 −0.123396
\(347\) −15.8265 −0.849610 −0.424805 0.905285i \(-0.639657\pi\)
−0.424805 + 0.905285i \(0.639657\pi\)
\(348\) 4.16822 0.223440
\(349\) 3.65930 0.195878 0.0979388 0.995192i \(-0.468775\pi\)
0.0979388 + 0.995192i \(0.468775\pi\)
\(350\) −4.98194 −0.266296
\(351\) −5.16724 −0.275807
\(352\) 0 0
\(353\) −27.4937 −1.46334 −0.731671 0.681658i \(-0.761259\pi\)
−0.731671 + 0.681658i \(0.761259\pi\)
\(354\) 9.14651 0.486131
\(355\) 9.98194 0.529786
\(356\) 1.66604 0.0882999
\(357\) −21.4342 −1.13442
\(358\) 2.35119 0.124265
\(359\) −0.478740 −0.0252670 −0.0126335 0.999920i \(-0.504021\pi\)
−0.0126335 + 0.999920i \(0.504021\pi\)
\(360\) −3.07901 −0.162278
\(361\) 12.3597 0.650512
\(362\) −23.4033 −1.23005
\(363\) 0 0
\(364\) 14.3852 0.753992
\(365\) 14.5625 0.762237
\(366\) −5.64579 −0.295110
\(367\) −25.4582 −1.32891 −0.664453 0.747330i \(-0.731335\pi\)
−0.664453 + 0.747330i \(0.731335\pi\)
\(368\) 0.501062 0.0261197
\(369\) −5.86100 −0.305111
\(370\) 4.56494 0.237320
\(371\) 10.8745 0.564574
\(372\) 1.83846 0.0953198
\(373\) 2.75967 0.142890 0.0714451 0.997445i \(-0.477239\pi\)
0.0714451 + 0.997445i \(0.477239\pi\)
\(374\) 0 0
\(375\) −1.00000 −0.0516398
\(376\) −0.732428 −0.0377721
\(377\) 33.1656 1.70811
\(378\) −4.98194 −0.256243
\(379\) −21.6535 −1.11227 −0.556133 0.831093i \(-0.687716\pi\)
−0.556133 + 0.831093i \(0.687716\pi\)
\(380\) −3.63670 −0.186559
\(381\) −1.53114 −0.0784428
\(382\) −14.7188 −0.753078
\(383\) 16.2295 0.829287 0.414643 0.909984i \(-0.363906\pi\)
0.414643 + 0.909984i \(0.363906\pi\)
\(384\) −3.01930 −0.154078
\(385\) 0 0
\(386\) −16.1912 −0.824109
\(387\) 8.90173 0.452500
\(388\) −1.30661 −0.0663332
\(389\) −16.2916 −0.826019 −0.413009 0.910727i \(-0.635522\pi\)
−0.413009 + 0.910727i \(0.635522\pi\)
\(390\) −6.00509 −0.304079
\(391\) 1.09909 0.0555836
\(392\) 35.0299 1.76928
\(393\) −2.66108 −0.134234
\(394\) 16.9744 0.855157
\(395\) 8.00194 0.402621
\(396\) 0 0
\(397\) −19.9673 −1.00213 −0.501064 0.865410i \(-0.667058\pi\)
−0.501064 + 0.865410i \(0.667058\pi\)
\(398\) 13.8003 0.691747
\(399\) −24.0062 −1.20181
\(400\) −2.27943 −0.113972
\(401\) −27.3269 −1.36464 −0.682320 0.731053i \(-0.739029\pi\)
−0.682320 + 0.731053i \(0.739029\pi\)
\(402\) −14.0945 −0.702969
\(403\) 14.6282 0.728683
\(404\) −4.92570 −0.245063
\(405\) −1.00000 −0.0496904
\(406\) 31.9763 1.58696
\(407\) 0 0
\(408\) −15.3950 −0.762168
\(409\) −24.9380 −1.23310 −0.616552 0.787314i \(-0.711471\pi\)
−0.616552 + 0.787314i \(0.711471\pi\)
\(410\) −6.81134 −0.336388
\(411\) 10.2555 0.505865
\(412\) 10.9547 0.539698
\(413\) −33.7389 −1.66019
\(414\) 0.255462 0.0125553
\(415\) 12.5530 0.616200
\(416\) 18.1317 0.888981
\(417\) 5.29507 0.259301
\(418\) 0 0
\(419\) 16.8256 0.821986 0.410993 0.911639i \(-0.365182\pi\)
0.410993 + 0.911639i \(0.365182\pi\)
\(420\) 2.78393 0.135842
\(421\) 23.6006 1.15022 0.575112 0.818074i \(-0.304958\pi\)
0.575112 + 0.818074i \(0.304958\pi\)
\(422\) −25.9761 −1.26450
\(423\) −0.237878 −0.0115660
\(424\) 7.81054 0.379313
\(425\) −5.00000 −0.242536
\(426\) 11.6005 0.562045
\(427\) 20.8258 1.00783
\(428\) 5.04050 0.243642
\(429\) 0 0
\(430\) 10.3451 0.498886
\(431\) 37.6556 1.81381 0.906903 0.421339i \(-0.138440\pi\)
0.906903 + 0.421339i \(0.138440\pi\)
\(432\) −2.27943 −0.109669
\(433\) −30.9752 −1.48857 −0.744286 0.667861i \(-0.767210\pi\)
−0.744286 + 0.667861i \(0.767210\pi\)
\(434\) 14.1037 0.676997
\(435\) 6.41843 0.307740
\(436\) 2.89425 0.138609
\(437\) 1.23098 0.0588858
\(438\) 16.9238 0.808650
\(439\) 28.2131 1.34654 0.673268 0.739399i \(-0.264890\pi\)
0.673268 + 0.739399i \(0.264890\pi\)
\(440\) 0 0
\(441\) 11.3770 0.541762
\(442\) −30.0254 −1.42816
\(443\) 14.6420 0.695663 0.347831 0.937557i \(-0.386918\pi\)
0.347831 + 0.937557i \(0.386918\pi\)
\(444\) −2.55091 −0.121061
\(445\) 2.56545 0.121614
\(446\) 22.2648 1.05427
\(447\) −3.78841 −0.179186
\(448\) 37.0247 1.74925
\(449\) 11.9977 0.566206 0.283103 0.959090i \(-0.408636\pi\)
0.283103 + 0.959090i \(0.408636\pi\)
\(450\) −1.16215 −0.0547841
\(451\) 0 0
\(452\) 5.00858 0.235584
\(453\) 5.92001 0.278146
\(454\) 17.8691 0.838636
\(455\) 22.1511 1.03846
\(456\) −17.2424 −0.807448
\(457\) −10.0437 −0.469824 −0.234912 0.972017i \(-0.575480\pi\)
−0.234912 + 0.972017i \(0.575480\pi\)
\(458\) −10.3446 −0.483370
\(459\) −5.00000 −0.233380
\(460\) −0.142753 −0.00665591
\(461\) 13.4491 0.626386 0.313193 0.949689i \(-0.398601\pi\)
0.313193 + 0.949689i \(0.398601\pi\)
\(462\) 0 0
\(463\) 18.7836 0.872946 0.436473 0.899717i \(-0.356227\pi\)
0.436473 + 0.899717i \(0.356227\pi\)
\(464\) 14.6304 0.679199
\(465\) 2.83095 0.131282
\(466\) 26.5390 1.22940
\(467\) 26.5510 1.22863 0.614316 0.789060i \(-0.289432\pi\)
0.614316 + 0.789060i \(0.289432\pi\)
\(468\) 3.35567 0.155116
\(469\) 51.9907 2.40071
\(470\) −0.276449 −0.0127516
\(471\) −7.42909 −0.342314
\(472\) −24.2329 −1.11541
\(473\) 0 0
\(474\) 9.29944 0.427137
\(475\) −5.59998 −0.256944
\(476\) 13.9197 0.638007
\(477\) 2.53671 0.116148
\(478\) 19.5944 0.896228
\(479\) −10.7830 −0.492689 −0.246345 0.969182i \(-0.579229\pi\)
−0.246345 + 0.969182i \(0.579229\pi\)
\(480\) 3.50898 0.160162
\(481\) −20.2970 −0.925464
\(482\) −13.9001 −0.633134
\(483\) −0.942328 −0.0428774
\(484\) 0 0
\(485\) −2.01199 −0.0913596
\(486\) −1.16215 −0.0527161
\(487\) 6.81095 0.308634 0.154317 0.988021i \(-0.450682\pi\)
0.154317 + 0.988021i \(0.450682\pi\)
\(488\) 14.9580 0.677119
\(489\) 5.32086 0.240617
\(490\) 13.2218 0.597298
\(491\) −26.0954 −1.17767 −0.588834 0.808254i \(-0.700413\pi\)
−0.588834 + 0.808254i \(0.700413\pi\)
\(492\) 3.80621 0.171597
\(493\) 32.0922 1.44536
\(494\) −33.6283 −1.51301
\(495\) 0 0
\(496\) 6.45297 0.289747
\(497\) −42.7910 −1.91944
\(498\) 14.5884 0.653721
\(499\) 19.3834 0.867719 0.433860 0.900981i \(-0.357151\pi\)
0.433860 + 0.900981i \(0.357151\pi\)
\(500\) 0.649414 0.0290427
\(501\) 1.97139 0.0880751
\(502\) −18.5983 −0.830084
\(503\) −18.8422 −0.840134 −0.420067 0.907493i \(-0.637993\pi\)
−0.420067 + 0.907493i \(0.637993\pi\)
\(504\) 13.1992 0.587940
\(505\) −7.58484 −0.337521
\(506\) 0 0
\(507\) 13.7003 0.608453
\(508\) 0.994345 0.0441169
\(509\) −5.15120 −0.228323 −0.114162 0.993462i \(-0.536418\pi\)
−0.114162 + 0.993462i \(0.536418\pi\)
\(510\) −5.81074 −0.257304
\(511\) −62.4272 −2.76162
\(512\) 22.0353 0.973831
\(513\) −5.59998 −0.247245
\(514\) −6.36212 −0.280621
\(515\) 16.8685 0.743317
\(516\) −5.78091 −0.254490
\(517\) 0 0
\(518\) −19.5692 −0.859820
\(519\) 1.97504 0.0866948
\(520\) 15.9100 0.697698
\(521\) −41.5022 −1.81824 −0.909121 0.416532i \(-0.863246\pi\)
−0.909121 + 0.416532i \(0.863246\pi\)
\(522\) 7.45917 0.326479
\(523\) 6.92280 0.302713 0.151356 0.988479i \(-0.451636\pi\)
0.151356 + 0.988479i \(0.451636\pi\)
\(524\) 1.72815 0.0754944
\(525\) 4.28684 0.187093
\(526\) 13.9273 0.607259
\(527\) 14.1548 0.616592
\(528\) 0 0
\(529\) −22.9517 −0.997899
\(530\) 2.94803 0.128054
\(531\) −7.87035 −0.341544
\(532\) 15.5900 0.675911
\(533\) 30.2852 1.31180
\(534\) 2.98143 0.129019
\(535\) 7.76161 0.335564
\(536\) 37.3421 1.61293
\(537\) −2.02315 −0.0873052
\(538\) −15.8592 −0.683740
\(539\) 0 0
\(540\) 0.649414 0.0279463
\(541\) 28.4183 1.22180 0.610899 0.791709i \(-0.290808\pi\)
0.610899 + 0.791709i \(0.290808\pi\)
\(542\) 13.7573 0.590928
\(543\) 20.1380 0.864205
\(544\) 17.5449 0.752231
\(545\) 4.45671 0.190905
\(546\) 25.7429 1.10169
\(547\) −25.8592 −1.10566 −0.552829 0.833295i \(-0.686452\pi\)
−0.552829 + 0.833295i \(0.686452\pi\)
\(548\) −6.66004 −0.284503
\(549\) 4.85807 0.207337
\(550\) 0 0
\(551\) 35.9431 1.53123
\(552\) −0.676824 −0.0288075
\(553\) −34.3031 −1.45871
\(554\) 11.9281 0.506776
\(555\) −3.92802 −0.166735
\(556\) −3.43869 −0.145833
\(557\) 8.06409 0.341687 0.170843 0.985298i \(-0.445351\pi\)
0.170843 + 0.985298i \(0.445351\pi\)
\(558\) 3.28999 0.139276
\(559\) −45.9973 −1.94548
\(560\) 9.77157 0.412924
\(561\) 0 0
\(562\) −3.41542 −0.144071
\(563\) −29.6516 −1.24966 −0.624832 0.780759i \(-0.714833\pi\)
−0.624832 + 0.780759i \(0.714833\pi\)
\(564\) 0.154481 0.00650483
\(565\) 7.71247 0.324466
\(566\) 25.3673 1.06627
\(567\) 4.28684 0.180030
\(568\) −30.7345 −1.28959
\(569\) 42.1026 1.76503 0.882517 0.470280i \(-0.155847\pi\)
0.882517 + 0.470280i \(0.155847\pi\)
\(570\) −6.50800 −0.272590
\(571\) 26.6823 1.11662 0.558311 0.829632i \(-0.311450\pi\)
0.558311 + 0.829632i \(0.311450\pi\)
\(572\) 0 0
\(573\) 12.6652 0.529094
\(574\) 29.1992 1.21875
\(575\) −0.219819 −0.00916708
\(576\) 8.63682 0.359867
\(577\) −3.30818 −0.137721 −0.0688606 0.997626i \(-0.521936\pi\)
−0.0688606 + 0.997626i \(0.521936\pi\)
\(578\) −9.29718 −0.386712
\(579\) 13.9321 0.578999
\(580\) −4.16822 −0.173076
\(581\) −53.8125 −2.23252
\(582\) −2.33822 −0.0969225
\(583\) 0 0
\(584\) −44.8381 −1.85542
\(585\) 5.16724 0.213639
\(586\) −28.7823 −1.18899
\(587\) −8.52770 −0.351976 −0.175988 0.984392i \(-0.556312\pi\)
−0.175988 + 0.984392i \(0.556312\pi\)
\(588\) −7.38839 −0.304692
\(589\) 15.8533 0.653223
\(590\) −9.14651 −0.376556
\(591\) −14.6060 −0.600813
\(592\) −8.95367 −0.367993
\(593\) 23.4343 0.962333 0.481167 0.876629i \(-0.340213\pi\)
0.481167 + 0.876629i \(0.340213\pi\)
\(594\) 0 0
\(595\) 21.4342 0.878717
\(596\) 2.46025 0.100776
\(597\) −11.8748 −0.486004
\(598\) −1.32003 −0.0539801
\(599\) 18.8344 0.769555 0.384777 0.923009i \(-0.374278\pi\)
0.384777 + 0.923009i \(0.374278\pi\)
\(600\) 3.07901 0.125700
\(601\) 37.6585 1.53612 0.768062 0.640376i \(-0.221221\pi\)
0.768062 + 0.640376i \(0.221221\pi\)
\(602\) −44.3479 −1.80749
\(603\) 12.1280 0.493889
\(604\) −3.84453 −0.156432
\(605\) 0 0
\(606\) −8.81471 −0.358073
\(607\) 29.3680 1.19201 0.596004 0.802981i \(-0.296754\pi\)
0.596004 + 0.802981i \(0.296754\pi\)
\(608\) 19.6502 0.796921
\(609\) −27.5148 −1.11496
\(610\) 5.64579 0.228592
\(611\) 1.22917 0.0497269
\(612\) 3.24707 0.131255
\(613\) 12.9111 0.521474 0.260737 0.965410i \(-0.416035\pi\)
0.260737 + 0.965410i \(0.416035\pi\)
\(614\) −5.74127 −0.231699
\(615\) 5.86100 0.236338
\(616\) 0 0
\(617\) 41.6041 1.67492 0.837459 0.546500i \(-0.184040\pi\)
0.837459 + 0.546500i \(0.184040\pi\)
\(618\) 19.6037 0.788578
\(619\) 4.27117 0.171673 0.0858363 0.996309i \(-0.472644\pi\)
0.0858363 + 0.996309i \(0.472644\pi\)
\(620\) −1.83846 −0.0738344
\(621\) −0.219819 −0.00882102
\(622\) 16.0488 0.643497
\(623\) −10.9977 −0.440613
\(624\) 11.7784 0.471512
\(625\) 1.00000 0.0400000
\(626\) 4.72918 0.189016
\(627\) 0 0
\(628\) 4.82455 0.192521
\(629\) −19.6401 −0.783103
\(630\) 4.98194 0.198485
\(631\) −27.5510 −1.09679 −0.548393 0.836221i \(-0.684760\pi\)
−0.548393 + 0.836221i \(0.684760\pi\)
\(632\) −24.6381 −0.980049
\(633\) 22.3518 0.888406
\(634\) −30.0066 −1.19171
\(635\) 1.53114 0.0607615
\(636\) −1.64737 −0.0653225
\(637\) −58.7877 −2.32925
\(638\) 0 0
\(639\) −9.98194 −0.394879
\(640\) 3.01930 0.119348
\(641\) 9.16806 0.362117 0.181058 0.983472i \(-0.442048\pi\)
0.181058 + 0.983472i \(0.442048\pi\)
\(642\) 9.02014 0.355996
\(643\) −3.40908 −0.134441 −0.0672206 0.997738i \(-0.521413\pi\)
−0.0672206 + 0.997738i \(0.521413\pi\)
\(644\) 0.611961 0.0241146
\(645\) −8.90173 −0.350505
\(646\) −32.5400 −1.28027
\(647\) 22.3608 0.879092 0.439546 0.898220i \(-0.355139\pi\)
0.439546 + 0.898220i \(0.355139\pi\)
\(648\) 3.07901 0.120955
\(649\) 0 0
\(650\) 6.00509 0.235539
\(651\) −12.1359 −0.475642
\(652\) −3.45544 −0.135325
\(653\) −14.7957 −0.578999 −0.289500 0.957178i \(-0.593489\pi\)
−0.289500 + 0.957178i \(0.593489\pi\)
\(654\) 5.17936 0.202529
\(655\) 2.66108 0.103977
\(656\) 13.3598 0.521611
\(657\) −14.5625 −0.568138
\(658\) 1.18509 0.0461997
\(659\) 47.4724 1.84926 0.924631 0.380864i \(-0.124373\pi\)
0.924631 + 0.380864i \(0.124373\pi\)
\(660\) 0 0
\(661\) −21.6525 −0.842184 −0.421092 0.907018i \(-0.638353\pi\)
−0.421092 + 0.907018i \(0.638353\pi\)
\(662\) 3.89328 0.151317
\(663\) 25.8362 1.00339
\(664\) −38.6507 −1.49994
\(665\) 24.0062 0.930921
\(666\) −4.56494 −0.176888
\(667\) 1.41089 0.0546300
\(668\) −1.28025 −0.0495342
\(669\) −19.1583 −0.740703
\(670\) 14.0945 0.544518
\(671\) 0 0
\(672\) −15.0424 −0.580275
\(673\) 38.3923 1.47991 0.739957 0.672654i \(-0.234846\pi\)
0.739957 + 0.672654i \(0.234846\pi\)
\(674\) −31.8426 −1.22653
\(675\) 1.00000 0.0384900
\(676\) −8.89718 −0.342199
\(677\) 22.9333 0.881398 0.440699 0.897655i \(-0.354731\pi\)
0.440699 + 0.897655i \(0.354731\pi\)
\(678\) 8.96302 0.344223
\(679\) 8.62507 0.331000
\(680\) 15.3950 0.590373
\(681\) −15.3759 −0.589206
\(682\) 0 0
\(683\) 25.9084 0.991359 0.495680 0.868505i \(-0.334919\pi\)
0.495680 + 0.868505i \(0.334919\pi\)
\(684\) 3.63670 0.139053
\(685\) −10.2555 −0.391841
\(686\) −21.8060 −0.832557
\(687\) 8.90126 0.339604
\(688\) −20.2909 −0.773584
\(689\) −13.1078 −0.499366
\(690\) −0.255462 −0.00972526
\(691\) 14.8155 0.563608 0.281804 0.959472i \(-0.409067\pi\)
0.281804 + 0.959472i \(0.409067\pi\)
\(692\) −1.28262 −0.0487579
\(693\) 0 0
\(694\) 18.3927 0.698177
\(695\) −5.29507 −0.200854
\(696\) −19.7624 −0.749092
\(697\) 29.3050 1.11001
\(698\) −4.25264 −0.160965
\(699\) −22.8362 −0.863744
\(700\) −2.78393 −0.105223
\(701\) −5.50632 −0.207971 −0.103985 0.994579i \(-0.533160\pi\)
−0.103985 + 0.994579i \(0.533160\pi\)
\(702\) 6.00509 0.226647
\(703\) −21.9968 −0.829626
\(704\) 0 0
\(705\) 0.237878 0.00895899
\(706\) 31.9517 1.20252
\(707\) 32.5150 1.22285
\(708\) 5.11112 0.192087
\(709\) −17.9007 −0.672274 −0.336137 0.941813i \(-0.609120\pi\)
−0.336137 + 0.941813i \(0.609120\pi\)
\(710\) −11.6005 −0.435358
\(711\) −8.00194 −0.300096
\(712\) −7.89905 −0.296029
\(713\) 0.622297 0.0233052
\(714\) 24.9097 0.932222
\(715\) 0 0
\(716\) 1.31386 0.0491012
\(717\) −16.8605 −0.629668
\(718\) 0.556367 0.0207634
\(719\) 32.5918 1.21547 0.607734 0.794141i \(-0.292079\pi\)
0.607734 + 0.794141i \(0.292079\pi\)
\(720\) 2.27943 0.0849495
\(721\) −72.3128 −2.69307
\(722\) −14.3638 −0.534566
\(723\) 11.9607 0.444825
\(724\) −13.0779 −0.486037
\(725\) −6.41843 −0.238375
\(726\) 0 0
\(727\) −43.0199 −1.59552 −0.797759 0.602976i \(-0.793982\pi\)
−0.797759 + 0.602976i \(0.793982\pi\)
\(728\) −68.2035 −2.52779
\(729\) 1.00000 0.0370370
\(730\) −16.9238 −0.626378
\(731\) −44.5087 −1.64621
\(732\) −3.15490 −0.116608
\(733\) −40.7066 −1.50353 −0.751766 0.659429i \(-0.770798\pi\)
−0.751766 + 0.659429i \(0.770798\pi\)
\(734\) 29.5862 1.09204
\(735\) −11.3770 −0.419647
\(736\) 0.771340 0.0284320
\(737\) 0 0
\(738\) 6.81134 0.250729
\(739\) −38.1240 −1.40241 −0.701207 0.712958i \(-0.747355\pi\)
−0.701207 + 0.712958i \(0.747355\pi\)
\(740\) 2.55091 0.0937734
\(741\) 28.9364 1.06300
\(742\) −12.6377 −0.463945
\(743\) 18.0381 0.661754 0.330877 0.943674i \(-0.392655\pi\)
0.330877 + 0.943674i \(0.392655\pi\)
\(744\) −8.71654 −0.319564
\(745\) 3.78841 0.138797
\(746\) −3.20714 −0.117422
\(747\) −12.5530 −0.459289
\(748\) 0 0
\(749\) −33.2728 −1.21576
\(750\) 1.16215 0.0424356
\(751\) −17.5349 −0.639859 −0.319929 0.947441i \(-0.603659\pi\)
−0.319929 + 0.947441i \(0.603659\pi\)
\(752\) 0.542227 0.0197730
\(753\) 16.0034 0.583197
\(754\) −38.5433 −1.40366
\(755\) −5.92001 −0.215451
\(756\) −2.78393 −0.101251
\(757\) −25.9609 −0.943565 −0.471782 0.881715i \(-0.656389\pi\)
−0.471782 + 0.881715i \(0.656389\pi\)
\(758\) 25.1646 0.914019
\(759\) 0 0
\(760\) 17.2424 0.625447
\(761\) −28.0059 −1.01521 −0.507607 0.861589i \(-0.669470\pi\)
−0.507607 + 0.861589i \(0.669470\pi\)
\(762\) 1.77941 0.0644613
\(763\) −19.1052 −0.691655
\(764\) −8.22493 −0.297567
\(765\) 5.00000 0.180775
\(766\) −18.8610 −0.681477
\(767\) 40.6680 1.46843
\(768\) −13.7648 −0.496693
\(769\) −22.9537 −0.827732 −0.413866 0.910338i \(-0.635822\pi\)
−0.413866 + 0.910338i \(0.635822\pi\)
\(770\) 0 0
\(771\) 5.47446 0.197158
\(772\) −9.04772 −0.325634
\(773\) −23.5282 −0.846251 −0.423126 0.906071i \(-0.639067\pi\)
−0.423126 + 0.906071i \(0.639067\pi\)
\(774\) −10.3451 −0.371848
\(775\) −2.83095 −0.101691
\(776\) 6.19492 0.222385
\(777\) 16.8388 0.604089
\(778\) 18.9333 0.678791
\(779\) 32.8215 1.17595
\(780\) −3.35567 −0.120152
\(781\) 0 0
\(782\) −1.27731 −0.0456765
\(783\) −6.41843 −0.229376
\(784\) −25.9331 −0.926184
\(785\) 7.42909 0.265155
\(786\) 3.09257 0.110308
\(787\) 25.2725 0.900869 0.450434 0.892810i \(-0.351269\pi\)
0.450434 + 0.892810i \(0.351269\pi\)
\(788\) 9.48537 0.337902
\(789\) −11.9841 −0.426645
\(790\) −9.29944 −0.330859
\(791\) −33.0621 −1.17555
\(792\) 0 0
\(793\) −25.1028 −0.891427
\(794\) 23.2049 0.823511
\(795\) −2.53671 −0.0899677
\(796\) 7.71168 0.273333
\(797\) 8.65645 0.306627 0.153314 0.988178i \(-0.451006\pi\)
0.153314 + 0.988178i \(0.451006\pi\)
\(798\) 27.8987 0.987605
\(799\) 1.18939 0.0420775
\(800\) −3.50898 −0.124061
\(801\) −2.56545 −0.0906457
\(802\) 31.7579 1.12141
\(803\) 0 0
\(804\) −7.87607 −0.277768
\(805\) 0.942328 0.0332127
\(806\) −17.0001 −0.598804
\(807\) 13.6465 0.480379
\(808\) 23.3538 0.821584
\(809\) 10.2124 0.359050 0.179525 0.983753i \(-0.442544\pi\)
0.179525 + 0.983753i \(0.442544\pi\)
\(810\) 1.16215 0.0408337
\(811\) 47.6224 1.67225 0.836124 0.548540i \(-0.184816\pi\)
0.836124 + 0.548540i \(0.184816\pi\)
\(812\) 17.8685 0.627061
\(813\) −11.8379 −0.415172
\(814\) 0 0
\(815\) −5.32086 −0.186381
\(816\) 11.3972 0.398981
\(817\) −49.8495 −1.74401
\(818\) 28.9816 1.01332
\(819\) −22.1511 −0.774023
\(820\) −3.80621 −0.132919
\(821\) 47.5598 1.65985 0.829925 0.557875i \(-0.188383\pi\)
0.829925 + 0.557875i \(0.188383\pi\)
\(822\) −11.9184 −0.415700
\(823\) −24.7360 −0.862242 −0.431121 0.902294i \(-0.641882\pi\)
−0.431121 + 0.902294i \(0.641882\pi\)
\(824\) −51.9384 −1.80936
\(825\) 0 0
\(826\) 39.2096 1.36428
\(827\) 1.56166 0.0543043 0.0271522 0.999631i \(-0.491356\pi\)
0.0271522 + 0.999631i \(0.491356\pi\)
\(828\) 0.142753 0.00496102
\(829\) 42.1950 1.46549 0.732747 0.680501i \(-0.238238\pi\)
0.732747 + 0.680501i \(0.238238\pi\)
\(830\) −14.5884 −0.506370
\(831\) −10.2638 −0.356048
\(832\) −44.6285 −1.54721
\(833\) −56.8851 −1.97095
\(834\) −6.15366 −0.213084
\(835\) −1.97139 −0.0682227
\(836\) 0 0
\(837\) −2.83095 −0.0978521
\(838\) −19.5539 −0.675477
\(839\) −6.29451 −0.217311 −0.108655 0.994079i \(-0.534654\pi\)
−0.108655 + 0.994079i \(0.534654\pi\)
\(840\) −13.1992 −0.455416
\(841\) 12.1963 0.420562
\(842\) −27.4274 −0.945211
\(843\) 2.93889 0.101221
\(844\) −14.5156 −0.499647
\(845\) −13.7003 −0.471305
\(846\) 0.276449 0.00950451
\(847\) 0 0
\(848\) −5.78225 −0.198563
\(849\) −21.8279 −0.749133
\(850\) 5.81074 0.199307
\(851\) −0.863453 −0.0295988
\(852\) 6.48241 0.222084
\(853\) 7.27120 0.248961 0.124481 0.992222i \(-0.460274\pi\)
0.124481 + 0.992222i \(0.460274\pi\)
\(854\) −24.2026 −0.828197
\(855\) 5.59998 0.191515
\(856\) −23.8981 −0.816819
\(857\) −39.8590 −1.36156 −0.680779 0.732489i \(-0.738359\pi\)
−0.680779 + 0.732489i \(0.738359\pi\)
\(858\) 0 0
\(859\) 13.5278 0.461564 0.230782 0.973006i \(-0.425872\pi\)
0.230782 + 0.973006i \(0.425872\pi\)
\(860\) 5.78091 0.197127
\(861\) −25.1252 −0.856264
\(862\) −43.7613 −1.49052
\(863\) 50.3117 1.71263 0.856316 0.516453i \(-0.172748\pi\)
0.856316 + 0.516453i \(0.172748\pi\)
\(864\) −3.50898 −0.119378
\(865\) −1.97504 −0.0671535
\(866\) 35.9977 1.22325
\(867\) 8.00000 0.271694
\(868\) 7.88119 0.267505
\(869\) 0 0
\(870\) −7.45917 −0.252889
\(871\) −62.6681 −2.12343
\(872\) −13.7223 −0.464694
\(873\) 2.01199 0.0680954
\(874\) −1.43058 −0.0483901
\(875\) −4.28684 −0.144922
\(876\) 9.45710 0.319526
\(877\) 1.20736 0.0407696 0.0203848 0.999792i \(-0.493511\pi\)
0.0203848 + 0.999792i \(0.493511\pi\)
\(878\) −32.7877 −1.10653
\(879\) 24.7665 0.835354
\(880\) 0 0
\(881\) 1.91816 0.0646245 0.0323123 0.999478i \(-0.489713\pi\)
0.0323123 + 0.999478i \(0.489713\pi\)
\(882\) −13.2218 −0.445200
\(883\) 58.0890 1.95485 0.977425 0.211281i \(-0.0677634\pi\)
0.977425 + 0.211281i \(0.0677634\pi\)
\(884\) −16.7784 −0.564318
\(885\) 7.87035 0.264559
\(886\) −17.0162 −0.571669
\(887\) −30.2427 −1.01545 −0.507725 0.861519i \(-0.669513\pi\)
−0.507725 + 0.861519i \(0.669513\pi\)
\(888\) 12.0944 0.405862
\(889\) −6.56377 −0.220142
\(890\) −2.98143 −0.0999378
\(891\) 0 0
\(892\) 12.4417 0.416578
\(893\) 1.33211 0.0445773
\(894\) 4.40269 0.147248
\(895\) 2.02315 0.0676263
\(896\) −12.9432 −0.432403
\(897\) 1.13586 0.0379251
\(898\) −13.9431 −0.465286
\(899\) 18.1703 0.606013
\(900\) −0.649414 −0.0216471
\(901\) −12.6835 −0.422550
\(902\) 0 0
\(903\) 38.1603 1.26990
\(904\) −23.7468 −0.789805
\(905\) −20.1380 −0.669410
\(906\) −6.87992 −0.228570
\(907\) −31.9669 −1.06144 −0.530721 0.847547i \(-0.678079\pi\)
−0.530721 + 0.847547i \(0.678079\pi\)
\(908\) 9.98532 0.331374
\(909\) 7.58484 0.251573
\(910\) −25.7429 −0.853368
\(911\) −51.8454 −1.71771 −0.858857 0.512216i \(-0.828825\pi\)
−0.858857 + 0.512216i \(0.828825\pi\)
\(912\) 12.7648 0.422684
\(913\) 0 0
\(914\) 11.6722 0.386083
\(915\) −4.85807 −0.160603
\(916\) −5.78060 −0.190996
\(917\) −11.4076 −0.376714
\(918\) 5.81074 0.191783
\(919\) 34.2084 1.12843 0.564215 0.825628i \(-0.309179\pi\)
0.564215 + 0.825628i \(0.309179\pi\)
\(920\) 0.676824 0.0223142
\(921\) 4.94023 0.162786
\(922\) −15.6298 −0.514741
\(923\) 51.5790 1.69774
\(924\) 0 0
\(925\) 3.92802 0.129153
\(926\) −21.8293 −0.717354
\(927\) −16.8685 −0.554036
\(928\) 22.5222 0.739326
\(929\) 18.3482 0.601984 0.300992 0.953627i \(-0.402682\pi\)
0.300992 + 0.953627i \(0.402682\pi\)
\(930\) −3.28999 −0.107883
\(931\) −63.7110 −2.08804
\(932\) 14.8301 0.485777
\(933\) −13.8096 −0.452106
\(934\) −30.8561 −1.00964
\(935\) 0 0
\(936\) −15.9100 −0.520033
\(937\) −3.38415 −0.110555 −0.0552777 0.998471i \(-0.517604\pi\)
−0.0552777 + 0.998471i \(0.517604\pi\)
\(938\) −60.4208 −1.97281
\(939\) −4.06934 −0.132798
\(940\) −0.154481 −0.00503862
\(941\) 35.1002 1.14423 0.572116 0.820172i \(-0.306123\pi\)
0.572116 + 0.820172i \(0.306123\pi\)
\(942\) 8.63369 0.281301
\(943\) 1.28836 0.0419547
\(944\) 17.9399 0.583895
\(945\) −4.28684 −0.139451
\(946\) 0 0
\(947\) −39.0512 −1.26899 −0.634497 0.772925i \(-0.718793\pi\)
−0.634497 + 0.772925i \(0.718793\pi\)
\(948\) 5.19657 0.168777
\(949\) 75.2480 2.44265
\(950\) 6.50800 0.211147
\(951\) 25.8199 0.837269
\(952\) −65.9961 −2.13895
\(953\) −24.6783 −0.799409 −0.399705 0.916644i \(-0.630887\pi\)
−0.399705 + 0.916644i \(0.630887\pi\)
\(954\) −2.94803 −0.0954458
\(955\) −12.6652 −0.409835
\(956\) 10.9495 0.354131
\(957\) 0 0
\(958\) 12.5315 0.404873
\(959\) 43.9635 1.41966
\(960\) −8.63682 −0.278752
\(961\) −22.9857 −0.741474
\(962\) 23.5881 0.760512
\(963\) −7.76161 −0.250114
\(964\) −7.76747 −0.250173
\(965\) −13.9321 −0.448491
\(966\) 1.09512 0.0352350
\(967\) −1.20724 −0.0388222 −0.0194111 0.999812i \(-0.506179\pi\)
−0.0194111 + 0.999812i \(0.506179\pi\)
\(968\) 0 0
\(969\) 27.9999 0.899486
\(970\) 2.33822 0.0750758
\(971\) 50.3821 1.61684 0.808418 0.588608i \(-0.200324\pi\)
0.808418 + 0.588608i \(0.200324\pi\)
\(972\) −0.649414 −0.0208300
\(973\) 22.6991 0.727701
\(974\) −7.91533 −0.253624
\(975\) −5.16724 −0.165484
\(976\) −11.0737 −0.354459
\(977\) −5.88330 −0.188223 −0.0941117 0.995562i \(-0.530001\pi\)
−0.0941117 + 0.995562i \(0.530001\pi\)
\(978\) −6.18362 −0.197730
\(979\) 0 0
\(980\) 7.38839 0.236013
\(981\) −4.45671 −0.142292
\(982\) 30.3267 0.967763
\(983\) 36.4899 1.16385 0.581924 0.813243i \(-0.302300\pi\)
0.581924 + 0.813243i \(0.302300\pi\)
\(984\) −18.0461 −0.575288
\(985\) 14.6060 0.465387
\(986\) −37.2958 −1.18774
\(987\) −1.01974 −0.0324588
\(988\) −18.7917 −0.597843
\(989\) −1.95677 −0.0622216
\(990\) 0 0
\(991\) −36.8404 −1.17027 −0.585137 0.810934i \(-0.698959\pi\)
−0.585137 + 0.810934i \(0.698959\pi\)
\(992\) 9.93376 0.315397
\(993\) −3.35008 −0.106311
\(994\) 49.7294 1.57732
\(995\) 11.8748 0.376457
\(996\) 8.15206 0.258308
\(997\) −34.8694 −1.10432 −0.552162 0.833737i \(-0.686197\pi\)
−0.552162 + 0.833737i \(0.686197\pi\)
\(998\) −22.5263 −0.713059
\(999\) 3.92802 0.124277
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.1.3 4
3.2 odd 2 5445.2.a.bv.1.2 4
5.4 even 2 9075.2.a.dj.1.2 4
11.7 odd 10 165.2.m.a.16.1 8
11.8 odd 10 165.2.m.a.31.1 yes 8
11.10 odd 2 1815.2.a.x.1.2 4
33.8 even 10 495.2.n.d.361.2 8
33.29 even 10 495.2.n.d.181.2 8
33.32 even 2 5445.2.a.be.1.3 4
55.7 even 20 825.2.bx.h.49.2 16
55.8 even 20 825.2.bx.h.724.2 16
55.18 even 20 825.2.bx.h.49.3 16
55.19 odd 10 825.2.n.k.526.2 8
55.29 odd 10 825.2.n.k.676.2 8
55.52 even 20 825.2.bx.h.724.3 16
55.54 odd 2 9075.2.a.cl.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
165.2.m.a.16.1 8 11.7 odd 10
165.2.m.a.31.1 yes 8 11.8 odd 10
495.2.n.d.181.2 8 33.29 even 10
495.2.n.d.361.2 8 33.8 even 10
825.2.n.k.526.2 8 55.19 odd 10
825.2.n.k.676.2 8 55.29 odd 10
825.2.bx.h.49.2 16 55.7 even 20
825.2.bx.h.49.3 16 55.18 even 20
825.2.bx.h.724.2 16 55.8 even 20
825.2.bx.h.724.3 16 55.52 even 20
1815.2.a.o.1.3 4 1.1 even 1 trivial
1815.2.a.x.1.2 4 11.10 odd 2
5445.2.a.be.1.3 4 33.32 even 2
5445.2.a.bv.1.2 4 3.2 odd 2
9075.2.a.cl.1.3 4 55.54 odd 2
9075.2.a.dj.1.2 4 5.4 even 2